{"id":73,"date":"2026-08-06T10:50:23","date_gmt":"2026-08-06T10:50:23","guid":{"rendered":"https:\/\/sqoolpapers.co.za\/notes\/algebra-equations-and-inequalities-g12-maths-t4p1\/"},"modified":"2026-08-22T16:27:39","modified_gmt":"2026-08-22T16:27:39","slug":"algebra-equations-and-inequalities-g12-maths-t4p1","status":"publish","type":"post","link":"https:\/\/repstaq.com\/study-guide-docs\/algebra-equations-and-inequalities-g12-maths-t4p1\/","title":{"rendered":"Algebra, Equations and Inequalities-G12-Maths-T4P1"},"content":{"rendered":"\n<div style=\"height:50px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<h2 class=\"wp-block-heading\">Algebra, Equations and Inequalities<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Algebra involves using symbols to represent unknown values and relationships. Solving an equation means finding every value of the unknown that makes the equation true. In examinations, show substitutions, factorisation and simplification clearly because marks are awarded for the method as well as the answer. Consistent accuracy is applied throughout the marking process.<\/p>\n\n\n\n<div style=\"height:50px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<h2 class=\"wp-block-heading\">Linear equations<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">A linear equation has an unknown raised only to the first power. Its general form is:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><span class=\"katex-eq\" data-katex-display=\"true\">\nax+b=0,\\quad a\\ne0\n<\/span><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">To solve a linear equation, collect terms containing the unknown on one side and constants on the other.<\/p>\n\n\n\n<div style=\"height:50px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<h3 class=\"wp-block-heading\">Example<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Question: Solve:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><span class=\"katex-eq\" data-katex-display=\"true\">\n3x+7=25\n<\/span><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Step 1: Identify the equation.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><span class=\"katex-eq\" data-katex-display=\"true\">\n3x+7=25\n<\/span><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Step 2: Subtract the constant from both sides.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><span class=\"katex-eq\" data-katex-display=\"true\">\n3x=25-7\n<\/span><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Step 3: Simplify.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><span class=\"katex-eq\" data-katex-display=\"true\">\n3x=18\n<\/span><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Step 4: Divide both sides by the coefficient.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><span class=\"katex-eq\" data-katex-display=\"true\">\nx=\\frac{18}{3}\n<\/span><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Step 5: State the final answer.<\/p>\n\n\n\n<div class=\"wp-block-math\"><math display=\"block\"><semantics><mrow><mi>x<\/mi><mo>=<\/mo><mn>6<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">x = 6<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<div style=\"height:50px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<h3 class=\"wp-block-heading\">Important<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Perform the same operation on both sides of the equation.<\/li>\n\n\n\n<li>Remove brackets before collecting like terms.<\/li>\n\n\n\n<li>Multiply through by the lowest common denominator when fractions occur.<\/li>\n\n\n\n<li>Check the answer by substituting it into the original equation.<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">Common Mistake: Moving a term across the equal sign without changing its sign correctly.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Practical Application: Linear equations can model costs, income, distance, mixtures and the number of items sold.<\/p>\n\n\n\n<div style=\"height:50px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<h2 class=\"wp-block-heading\">Quadratic equations<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">A quadratic equation has the general form:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><span class=\"katex-eq\" data-katex-display=\"true\">\nax^2+bx+c=0,\\quad a\\ne0\n<\/span><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">It can be solved by factorisation, completing the square or using the quadratic formula:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><span class=\"katex-eq\" data-katex-display=\"true\">\nx=\\frac{-b\\pm\\sqrt{b^2-4ac}}{2a}\n<\/span><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The expression under the square root is the discriminant:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><span class=\"katex-eq\" data-katex-display=\"true\">\n\\Delta=b^2-4ac\n<\/span><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>If the discriminant is positive, there are two unequal real roots.<\/li>\n\n\n\n<li>If it is zero, there are two equal real roots.<\/li>\n\n\n\n<li>If it is negative, there are no real roots.<\/li>\n<\/ul>\n\n\n\n<div style=\"height:50px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<h3 class=\"wp-block-heading\">Example<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Question: Solve:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><span class=\"katex-eq\" data-katex-display=\"true\">\n5x^2+2x=-9\n<\/span><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Step 1: Write the equation in standard form.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><span class=\"katex-eq\" data-katex-display=\"true\">\n5x^2+2x+9=0\n<\/span><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Step 2: Identify the coefficients.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><span class=\"katex-eq\" data-katex-display=\"true\">\na=5,\\quad b=2,\\quad c=9\n<\/span><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Step 3: Substitute into the quadratic formula.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><span class=\"katex-eq\" data-katex-display=\"true\">\nx=\\frac{-2\\pm\\sqrt{2^2-4(5)(9)}}{2(5)}\n<\/span><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Step 4: Simplify.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><span class=\"katex-eq\" data-katex-display=\"true\">\nx=\\frac{-2\\pm\\sqrt{-176}}{10}\n<\/span><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Step 5: Interpret the result.<\/p>\n\n\n\n<div class=\"wp-block-math\"><math display=\"block\"><semantics><mrow><mrow><mi mathvariant=\"normal\">\u0394<\/mi><\/mrow><mo>=<\/mo><mo form=\"prefix\" stretchy=\"false\">\u2212<\/mo><mn>176<\/mn><mo>&lt;<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">\u0394 = -176 &lt; 0<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<p class=\"wp-block-paragraph\"><span class=\"katex-eq\" data-katex-display=\"true\">\n\\therefore\\text{ there are no real roots}\n<\/span><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Exam Tip: Always place the equation in standard form before identifying the values of the coefficients.<\/p>\n\n\n\n<div style=\"height:50px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<figure class=\"wp-block-image size-large\"><a href=\"https:\/\/sqooltutors.co.za\/sign-up-b\/\"><img decoding=\"async\" src=\"https:\/\/lmxddlwowqlsyucleuyv.supabase.co\/storage\/v1\/object\/public\/pdf_ads\/batch%201\/Doc%20Image%20Aug%2022,%202026,%2004_21_26%20PM%20(1).png\" alt=\"\"\/><\/a><\/figure>\n\n\n\n<h2 class=\"wp-block-heading\">Equations requiring factorisation<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Factorisation rewrites an algebraic expression as a product. The zero-product property states:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><span class=\"katex-eq\" data-katex-display=\"true\">\nAB=0\\Rightarrow A=0\\text{ or }B=0\n<\/span><\/p>\n\n\n\n<div style=\"height:50px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<h3 class=\"wp-block-heading\">Example<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Question: Solve:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><span class=\"katex-eq\" data-katex-display=\"true\">\nx^2+3x-10=0\n<\/span><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Step 1: Find two numbers whose product is negative ten and whose sum is three.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><span class=\"katex-eq\" data-katex-display=\"true\">\n5(-2)=-10\n<\/span><\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><span class=\"katex-eq\" data-katex-display=\"true\">\n5+(-2)=3\n<\/span><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Step 2: Factorise.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><span class=\"katex-eq\" data-katex-display=\"true\">\n(x+5)(x-2)=0\n<\/span><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Step 3: Equate each factor to zero.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><span class=\"katex-eq\" data-katex-display=\"true\">\nx+5=0\n<\/span><\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><span class=\"katex-eq\" data-katex-display=\"true\">\nx-2=0\n<\/span><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Step 4: Solve each equation.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><span class=\"katex-eq\" data-katex-display=\"true\">\nx=-5\n<\/span><\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><span class=\"katex-eq\" data-katex-display=\"true\">\nx=2\n<\/span><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Step 5: State both answers.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><span class=\"katex-eq\" data-katex-display=\"true\">\nx=-5\\text{ or }x=2\n<\/span><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">This factorisation and zero-product approach is consistent with the method used to solve equations in the marking guidance.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Inequalities requiring factorisation follow a similar process, but the final answer is an interval.<\/p>\n\n\n\n<div style=\"height:50px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<h3 class=\"wp-block-heading\">Example<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\"><span class=\"katex-eq\" data-katex-display=\"true\">\n8x^2&gt;2x\n<\/span><\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><span class=\"katex-eq\" data-katex-display=\"true\">\n8x^2-2x&gt;0\n<\/span><\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><span class=\"katex-eq\" data-katex-display=\"true\">\n2x(4x-1)&gt;0\n<\/span><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The critical values are:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><span class=\"katex-eq\" data-katex-display=\"true\">\nx=0\\quad\\text{and}\\quad x=\\frac14\n<\/span><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">A sign analysis gives:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><span class=\"katex-eq\" data-katex-display=\"true\">\nx&lt;0\\quad\\text{or}\\quad x&gt;\\frac14\n<\/span><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Common Mistake: Dividing an inequality by an expression containing the unknown. Its sign may be unknown. Factorise and use critical values instead.<\/p>\n\n\n\n<div style=\"height:50px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<h2 class=\"wp-block-heading\">Exponential equations<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">An exponential equation contains the unknown in an exponent. Try to express both sides using the same base. If this is impossible, logarithms may be used.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Remember:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><span class=\"katex-eq\" data-katex-display=\"true\">\na^m=a^n\\Rightarrow m=n,\\quad a&gt;0,\\quad a\\ne1\n<\/span><\/p>\n\n\n\n<div style=\"height:50px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<h3 class=\"wp-block-heading\">Example<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Question: Solve:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><span class=\"katex-eq\" data-katex-display=\"true\">\n2(2^{2x})-9(2^x)+4=0\n<\/span><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Step 1: Use a substitution.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><span class=\"katex-eq\" data-katex-display=\"true\">\nk=2^x\n<\/span><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Step 2: Rewrite the equation.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><span class=\"katex-eq\" data-katex-display=\"true\">\n2k^2-9k+4=0\n<\/span><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Step 3: Factorise.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><span class=\"katex-eq\" data-katex-display=\"true\">\n(2k-1)(k-4)=0\n<\/span><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Step 4: Solve for the substituted variable.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><span class=\"katex-eq\" data-katex-display=\"true\">\nk=\\frac12\\quad\\text{or}\\quad k=4\n<\/span><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Step 5: Replace the substituted variable.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><span class=\"katex-eq\" data-katex-display=\"true\">\n2^x=\\frac12\\quad\\text{or}\\quad2^x=4\n<\/span><\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><span class=\"katex-eq\" data-katex-display=\"true\">\n2^x=2^{-1}\\quad\\text{or}\\quad2^x=2^2\n<\/span><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Step 6: State the final answer.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><span class=\"katex-eq\" data-katex-display=\"true\">\nx=-1\\quad\\text{or}\\quad x=2\n<\/span><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The substitution and factorisation method shown here follows the worked exponential equation solution.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Common Mistake: Forgetting to return to the original variable after solving the quadratic equation in the substituted variable.<\/p>\n\n\n\n<div style=\"height:50px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<h2 class=\"wp-block-heading\">Surd equations<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">A surd equation contains the unknown inside a square root. Isolate the surd before squaring. Squaring can introduce an extraneous solution, so every answer must be checked in the original equation.<\/p>\n\n\n\n<div style=\"height:50px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<h3 class=\"wp-block-heading\">Example<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Question: Solve:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><span class=\"katex-eq\" data-katex-display=\"true\">\n\\frac{1}{\\sqrt{x}}+\\frac{1}{x}=2\n<\/span><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Step 1: Isolate the surd term.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><span class=\"katex-eq\" data-katex-display=\"true\">\n\\frac{1}{\\sqrt{x}}=2-\\frac{1}{x}\n<\/span><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Step 2: Square both sides.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><span class=\"katex-eq\" data-katex-display=\"true\">\n\\frac{1}{x}=\\left(2-\\frac{1}{x}\\right)^2\n<\/span><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Step 3: Expand and simplify.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><span class=\"katex-eq\" data-katex-display=\"true\">\n\\frac{1}{x}=4-\\frac{4}{x}+\\frac{1}{x^2}\n<\/span><\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><span class=\"katex-eq\" data-katex-display=\"true\">\nx=4x^2-4x+1\n<\/span><\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><span class=\"katex-eq\" data-katex-display=\"true\">\n4x^2-5x+1=0\n<\/span><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Step 4: Factorise.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><span class=\"katex-eq\" data-katex-display=\"true\">\n(4x-1)(x-1)=0\n<\/span><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Step 5: Find the possible solutions.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><span class=\"katex-eq\" data-katex-display=\"true\">\nx=\\frac14\\quad\\text{or}\\quad x=1\n<\/span><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Step 6: Check in the original equation.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">For the first possible solution:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><span class=\"katex-eq\" data-katex-display=\"true\">\n\\frac{1}{\\sqrt{\\frac14}}+\\frac{1}{\\frac14}=2+4=6\n<\/span><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Therefore, it is rejected.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">For the second possible solution:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><span class=\"katex-eq\" data-katex-display=\"true\">\n\\frac{1}{\\sqrt{1}}+\\frac{1}{1}=1+1=2\n<\/span><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Step 7: State the final answer.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><span class=\"katex-eq\" data-katex-display=\"true\">\nx=1\n<\/span><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The selection of the valid root after checking is essential in surd equations.<\/p>\n\n\n\n<div style=\"height:50px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<h2 class=\"wp-block-heading\">Simultaneous equations<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Simultaneous equations must be satisfied by the same values of the unknowns. Two linear equations may be solved by elimination or substitution.<\/p>\n\n\n\n<div style=\"height:50px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<h3 class=\"wp-block-heading\">Example<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Question: Solve simultaneously:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><span class=\"katex-eq\" data-katex-display=\"true\">\nx+y=7\n<\/span><\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><span class=\"katex-eq\" data-katex-display=\"true\">\n2x-y=5\n<\/span><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Step 1: Add the equations to eliminate one unknown.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><span class=\"katex-eq\" data-katex-display=\"true\">\n(x+y)+(2x-y)=7+5\n<\/span><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Step 2: Simplify.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><span class=\"katex-eq\" data-katex-display=\"true\">\n3x=12\n<\/span><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Step 3: Solve for the first unknown.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><span class=\"katex-eq\" data-katex-display=\"true\">\nx=4\n<\/span><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Step 4: Substitute into the first equation.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><span class=\"katex-eq\" data-katex-display=\"true\">\n4+y=7\n<\/span><\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><span class=\"katex-eq\" data-katex-display=\"true\">\ny=3\n<\/span><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Step 5: State the ordered solution.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><span class=\"katex-eq\" data-katex-display=\"true\">\n(x;y)=(4;3)\n<\/span><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Exam Tip: Choose elimination when coefficients are equal or easy to make equal. Choose substitution when one unknown is already isolated.<\/p>\n\n\n\n<div style=\"height:50px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<figure class=\"wp-block-image size-large\"><a href=\"https:\/\/sqooltutors.co.za\/sign-up-b\/\"><img decoding=\"async\" src=\"https:\/\/lmxddlwowqlsyucleuyv.supabase.co\/storage\/v1\/object\/public\/pdf_ads\/batch%201\/Doc%20Image%20Aug%2022,%202026,%2004_21_31%20PM%20(1).png\" alt=\"\"\/><\/a><\/figure>\n\n\n\n<h2 class=\"wp-block-heading\">Linear and quadratic simultaneous equations<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">These consist of one linear equation and one quadratic equation. Use the linear equation to express one unknown in terms of the other, substitute into the quadratic equation and solve.<\/p>\n\n\n\n<div style=\"height:50px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<h3 class=\"wp-block-heading\">Example<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Question: Solve simultaneously:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><span class=\"katex-eq\" data-katex-display=\"true\">\nx=y+2\n<\/span><\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><span class=\"katex-eq\" data-katex-display=\"true\">\n5xy=x^2+6\n<\/span><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Step 1: Substitute the linear equation into the quadratic equation.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><span class=\"katex-eq\" data-katex-display=\"true\">\n5(y+2)y=(y+2)^2+6\n<\/span><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Step 2: Expand.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><span class=\"katex-eq\" data-katex-display=\"true\">\n5y^2+10y=y^2+4y+4+6\n<\/span><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Step 3: Write in standard form.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><span class=\"katex-eq\" data-katex-display=\"true\">\n4y^2+6y-10=0\n<\/span><\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><span class=\"katex-eq\" data-katex-display=\"true\">\n2y^2+3y-5=0\n<\/span><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Step 4: Factorise.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><span class=\"katex-eq\" data-katex-display=\"true\">\n(2y+5)(y-1)=0\n<\/span><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Step 5: Solve for the second unknown.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><span class=\"katex-eq\" data-katex-display=\"true\">\ny=-\\frac52\\quad\\text{or}\\quad y=1\n<\/span><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Step 6: Calculate the corresponding values of the first unknown.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><span class=\"katex-eq\" data-katex-display=\"true\">\nx=-\\frac52+2=-\\frac12\n<\/span><\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><span class=\"katex-eq\" data-katex-display=\"true\">\nx=1+2=3\n<\/span><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Step 7: State both ordered solutions.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><span class=\"katex-eq\" data-katex-display=\"true\">\n(x;y)=\\left(-\\frac12;-\\frac52\\right)\\quad\\text{or}\\quad(x;y)=(3;1)\n<\/span><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">This method and these ordered solutions correspond to the simultaneous-equation working provided.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Common Mistake: Matching an answer for one unknown with the incorrect corresponding answer for the other unknown.<\/p>\n\n\n\n<div style=\"height:50px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<h2 class=\"wp-block-heading\">Solving equations algebraically<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">An algebraic solution must show logical symbolic steps rather than relying only on a calculator or graph.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">A reliable approach is:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Determine the type of equation.<\/li>\n\n\n\n<li>State restrictions, especially for denominators and square roots.<\/li>\n\n\n\n<li>Simplify brackets, fractions and like terms.<\/li>\n\n\n\n<li>Write polynomial equations in standard form.<\/li>\n\n\n\n<li>Apply factorisation, substitution or an appropriate formula.<\/li>\n\n\n\n<li>Check possible extraneous solutions.<\/li>\n\n\n\n<li>State every valid solution clearly.<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">For rational equations, exclude values that make a denominator zero:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><span class=\"katex-eq\" data-katex-display=\"true\">\n\\frac{x+1}{x-3}=2,\\quad x\\ne3\n<\/span><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">For quadratic inequalities, find the critical values and use a sign diagram or number line. Do not include critical values when the inequality is strict.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Exam Tip: A calculator answer without the required algebraic steps may not earn all the available marks.<\/p>\n\n\n\n<div style=\"height:50px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<h2 class=\"wp-block-heading\">Rounding solutions to required accuracy<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Exact answers should normally be retained during calculations. Round only the final answer unless instructed otherwise.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Common instructions include:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Correct to two decimal places.<\/li>\n\n\n\n<li>Correct to the nearest integer.<\/li>\n\n\n\n<li>Correct to one decimal place.<\/li>\n\n\n\n<li>Correct to a specified number of significant figures.<\/li>\n<\/ul>\n\n\n\n<div style=\"height:50px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<h3 class=\"wp-block-heading\">Example<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Question: Solve correct to two decimal places:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><span class=\"katex-eq\" data-katex-display=\"true\">\n5x^2+9x+2=0\n<\/span><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Step 1: Identify the coefficients.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><span class=\"katex-eq\" data-katex-display=\"true\">\na=5,\\quad b=9,\\quad c=2\n<\/span><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Step 2: Substitute into the quadratic formula.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><span class=\"katex-eq\" data-katex-display=\"true\">\nx=\\frac{-9\\pm\\sqrt{9^2-4(5)(2)}}{2(5)}\n<\/span><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Step 3: Simplify.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><span class=\"katex-eq\" data-katex-display=\"true\">\nx=\\frac{-9\\pm\\sqrt{41}}{10}\n<\/span><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Step 4: Use a calculator.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><span class=\"katex-eq\" data-katex-display=\"true\">\nx=-0{,}259687\\ldots\\quad\\text{or}\\quad x=-1{,}540312\\ldots\n<\/span><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Step 5: Round to two decimal places.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><span class=\"katex-eq\" data-katex-display=\"true\">\nx=-0{,}26\\quad\\text{or}\\quad x=-1{,}54\n<\/span><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">These rounded values agree with the quadratic solutions shown in the marking guidance.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Remember: Do not round intermediate answers unnecessarily. Early rounding can cause the final answer to fall outside the accepted examination range.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Algebra, Equations and Inequalities Algebra involves using symbols to represent unknown values and relationships. Solving an equation means finding every value of the unknown that makes the equation true. In examinations, show substitutions, factorisation and simplification clearly because marks are awarded for the method as well as the answer. Consistent accuracy is applied throughout the [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"_breakdance_hide_in_design_set":false,"_breakdance_tags":"","footnotes":""},"categories":[1],"tags":[],"class_list":["post-73","post","type-post","status-publish","format-standard","hentry","category-uncategorized"],"acf":{"document_name":"Algebra, Equations and Inequalities","grade":12,"subject":"Mathematics","term":4,"paper":1},"_links":{"self":[{"href":"https:\/\/repstaq.com\/study-guide-docs\/wp-json\/wp\/v2\/posts\/73","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/repstaq.com\/study-guide-docs\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/repstaq.com\/study-guide-docs\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/repstaq.com\/study-guide-docs\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/repstaq.com\/study-guide-docs\/wp-json\/wp\/v2\/comments?post=73"}],"version-history":[{"count":5,"href":"https:\/\/repstaq.com\/study-guide-docs\/wp-json\/wp\/v2\/posts\/73\/revisions"}],"predecessor-version":[{"id":230,"href":"https:\/\/repstaq.com\/study-guide-docs\/wp-json\/wp\/v2\/posts\/73\/revisions\/230"}],"wp:attachment":[{"href":"https:\/\/repstaq.com\/study-guide-docs\/wp-json\/wp\/v2\/media?parent=73"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/repstaq.com\/study-guide-docs\/wp-json\/wp\/v2\/categories?post=73"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/repstaq.com\/study-guide-docs\/wp-json\/wp\/v2\/tags?post=73"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}