Solving Linear Equations Revision Notes for Grade 10 NSC Matric Mathematics

Solving Linear Equations

What is a linear equation?

A linear equation is an equation where the highest power (exponent) of the variable is 1. These are the simplest type of equations to solve and appear frequently in mathematics.

Examples of linear equations:

$2x + 2 = 1$
$\frac{2-x}{3x+1} = 2$
$4(2x - 9) - 4x = 4 - 6x$
$\frac{2a-3}{3} - 3a = \frac{a}{3}$

Key concepts you need to understand

Solving an equation means finding the value of the variable that makes the equation true. When you substitute this value back into the original equation, the left-hand side (LHS) equals the right-hand side (RHS).

The solution (also called the root) of an equation is the value of the variable that satisfies the equation. For linear equations, there is at most one solution.

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For example, to solve $x + 1 = 1$ , we need to find the value of $x$ that makes the LHS equal to the RHS. The solution is $x = 0$ .

The golden rule of solving equations

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An equation must always be balanced. Whatever operation you perform on the left-hand side, you must also perform on the right-hand side. This maintains the equality.

Method for solving linear equations

Follow these six steps systematically to solve any linear equation:

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The Six-Step Method:

Expand all brackets
Rearrange the terms so all variable terms are on one side and all constant terms are on the other side
Group like terms together and simplify
Factorise if necessary
Find the solution and write down the answer
Check the answer by substituting the solution back into the original equation

Worked example 1: Basic linear equation

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Worked Example: Basic Linear Equation

Question: Solve for $x$ : $4(2x - 9) - 4x = 4 - 6x$

Solution:

Step 1: Expand the brackets and simplify

$4(2x - 9) - 4x = 4 - 6x$

$8x - 36 - 4x = 4 - 6x$

$8x - 4x + 6x = 4 + 36$

$10x = 40$

Step 2: Divide both sides by 10

$x = 4$

Step 3: Check the answer

Substitute $x = 4$ back into the original equation:

LHS: $4[2(4) - 9] - 4(4) = 4(8 - 9) - 16 = 4(-1) - 16 = -4 - 16 = -20$
RHS: $4 - 6(4) = 4 - 24 = -20$

Since LHS = RHS, the answer is correct.

Worked example 2: Equation with fractions

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Worked Example: Equation with Fractions

Question: Solve for $x$ : $\frac{2-x}{3x+1} = 2$

Solution:

Step 1: Multiply both sides by $(3x + 1)$ Note: We need the restriction $x \neq -\frac{1}{3}$ to avoid division by zero.

$\frac{2-x}{3x+1} = 2$

$(2 - x) = 2(3x + 1)$

Step 2: Expand the brackets and simplify

$2 - x = 6x + 2$

$-x - 6x = 2 - 2$

$-7x = 0$

Step 3: Divide both sides by -7

$x = \frac{0}{-7} = 0$

Step 4: Check the answer

Substitute $x = 0$ back into the original equation:

LHS: $\frac{2-(0)}{3(0)+1} = \frac{2}{1} = 2$ = RHS

Since both sides are equal, the answer is correct.

Worked example 3: Complex equation with fractions

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Worked Example: Complex Equation with Fractions

Question: Solve for $a$ : $\frac{2a-3}{3} - 3a = \frac{a}{3}$

Solution:

Step 1: Multiply the equation by the common denominator 3

$2a - 3 - 9a = a$

$-7a - 3 = a$

Step 2: Rearrange the terms and simplify

$-7a - a = 3$

$-8a = 3$

Step 3: Divide both sides by -8

$a = -\frac{3}{8}$

Step 4: Check the answer by substituting back

LHS	RHS
$\displaystyle \frac{2\left(-\frac{3}{8}\right)-3}{3}\;-\;3\left(-\frac{3}{8}\right)$	$\displaystyle \frac{-\frac{3}{8}}{3}$
$\displaystyle \frac{-\frac{3}{4}-\frac{12}{4}}{3} + \frac{9}{8}$	$\displaystyle \frac{-\frac{3}{8}}{3}$
$\displaystyle \left(-\frac{15}{4} \times \frac{1}{3}\right) + \frac{9}{8}$	$\displaystyle -\frac{3}{8} \times \frac{1}{3}$
$\displaystyle -\frac{5}{4} + \frac{9}{8}$	$\displaystyle -\frac{1}{8}$
$\displaystyle -\frac{10}{8} + \frac{9}{8}$
$\displaystyle -\frac{1}{8}$	Final answer: $-\frac{1}{8}$

The verification shows that both LHS and RHS equal $-\frac{1}{8}$ , confirming our solution is correct.

Exam tips for solving linear equations

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Essential Exam Tips:

Always write each step clearly - examiners award marks for method
Remember to check your answer by substitution
When dealing with fractions, multiply through by the common denominator first
Watch out for restrictions (values that make denominators zero)
Keep your working neat and organised
If you get a fraction as an answer, leave it in simplest form unless asked to convert to decimal

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Key Points to Remember:

Linear equations have the highest power of the variable equal to 1
Always balance equations - what you do to one side, do to the other
Follow the six-step method systematically: expand, rearrange, group, factorise, solve, check
Check your answer by substituting back into the original equation
Linear equations have at most one solution - if you get more than one answer, check your work

Solving Linear Equations (Grade 10 NSC Matric Mathematics): Revision Notes