Solving Equations

Spec reference: Algebra - Solving equations and inequalities
Key idea: Solve linear and quadratic equations using algebraic and graphical methods.

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Solving linear equations

Use inverse operations to isolate the unknown. Whatever you do to one side, do to the other.

Example

Solve $3x+7=22$

$3x=22-7=15$

$x=15÷3=\boxed{{\displaystyle 5}}$

Example

Solve $5(2x-3)=25$

$2x-3=5$

$2x=8$

$x=\boxed{{\displaystyle 4}}$

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Equations with unknowns on both sides

Move all $x$ terms to one side.

Example

Solve $5x+3=2x+12$

$5x-2x=12-3$

$3x=9$

$x=\boxed{{\displaystyle 3}}$

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Equations involving fractions

Multiply through by the denominator to clear fractions.

Example

Solve ${\displaystyle \frac{x+3}{4}}=5$

$x+3=20$

$x=\boxed{{\displaystyle 17}}$

Example

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

Multiply through by 6 (LCM of 3 and 2):

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

$4x-2=3x+6$

$x=\boxed{{\displaystyle 8}}$

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Solving quadratic equations by factorising

If $AB=0$ then $A=0$ or $B=0$.

Example

Solve $x^2+5x+6=0$

Factorise: $(x+2)(x+3)=0$

$x+2=0$ → $x=-2$

$x+3=0$ → $x=-3$

$$\boxed{{\displaystyle x=-2\text{ or }x=-3}}$$

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The quadratic formula

For $a{x}^2+bx+c=0$:

$$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$$

Example

Solve $2x^2+5x-3=0$

$a=2,b=5,c=-3$

$$x=\frac{-5\pm \sqrt{25+24}}{4}=\frac{-5\pm 7}{4}$$

$x={\displaystyle \frac{2}{4}}={\displaystyle \frac{1}{2}}$ or $x={\displaystyle \frac{-12}{4}}=-3$

$$\boxed{{\displaystyle x=\frac{1}{2}\text{ or }x=-3}}$$

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Forming and solving equations

Example

The perimeter of a rectangle is 34 cm. The length is $(2x+1)$ cm and the width is $(x-2)$ cm. Find $x$.

$2(2x+1)+2(x-2)=34$

$4x+2+2x-4=34$

$6x-2=34$

$6x=36$

$x=\boxed{{\displaystyle 6}}$

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Exam tips

Watch out for

• Quadratic equations always equal zero before factorising or using the formula
• $x^2=9$ has two solutions: $x=3$ and $x=-3$
• Always check your answer by substituting back in

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Test Yourself

Question 1 of 5

Solve $3x + 7 = 22$