Year 10 Maths - 10A Linear Relations

Solving Linear Equations

Q: How do you solve an equation with x on both sides?

Expand any brackets, then add or subtract an x term on both sides so the pronumeral is on one side only. Then solve as a two-step equation.

20 practice questions 3 video lessons Theory + worked examples

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Practice questions

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Solving Linear Equations - Video - How to Solve Linear Equations With Variables on Both Sides Watch
Solving Linear Equations - Video - Strategies to Solve Multi Step Linear Equations with Fractions Watch
Solving Linear Equations - Video - Age word problems 1 | Linear equations Watch

Theory

To solve a linear equation, do the same thing to both sides until the pronumeral is on its own. This Year 10A maths page covers brackets, nested brackets, pronumerals on both sides, equations with fractions, checking by substitution, and writing and solving equations for word problems such as costs, ages, ratios, means and lengths.

An equation says two expressions are equal. A linear equation has the pronumeral to the power \(1\) only, such as \(5x - 3 = 2x + 9\). The solution is the value that makes both sides equal.

Solve by using inverse operations: \(+\) undoes \(-\), and \(\times\) undoes \(\div\). Do the same operation to both sides, so the equation stays balanced.

With brackets, expand first. With the pronumeral on both sides, move all pronumeral terms to one side first.

With fractions, multiply every term on both sides by the lowest common denominator. This clears the fractions in one step.

The answer can be a whole number, a fraction or a decimal, and it can be negative. Check it by substituting into the original equation: both sides must give the same number.

An equation is a balance. Whatever you do to one side, do to the other, and it stays level.

Solve by undoing the operations on \(x\) in reverse order, using the opposite operation each time.

Undo each operation with its inverse, on both sides:

\[ax + b = c \;\Rightarrow\; ax = c - b \;\Rightarrow\; x = \dfrac{c - b}{a}\]

x = \frac{c - b}{a}

Clearing a fraction: multiply both sides by the denominator.

\[\dfrac{ax + b}{d} = k \;\Rightarrow\; ax + b = dk\]

\frac{a x + b}{d} = k

One fraction on each side. Cross-multiply, keeping brackets: \(\dfrac{y + 2}{5} = \dfrac{y - 4}{2}\) gives \(2(y + 2) = 5(y - 4)\), so \(2y + 4 = 5y - 20\) and \(y = 8\).

Writing an equation from words. Let a pronumeral stand for the unknown, with units. $30 to join plus $12 a month, with $234 paid in total, gives \(30 + 12m = 234\), so \(m = 17\) months. For ages, write each person's age now in terms of one pronumeral, then add the same number of years to each.

Ratios. Two numbers in the ratio \(5 : 2\) can be written \(5k\) and \(2k\). If their sum is \(84\), then \(7k = 84\), so \(k = 12\) and the numbers are \(60\) and \(24\).

Means. If the mean of four numbers is \(15\), their total is \(4 \times 15 = 60\). With three of them \(12\), \(18\) and \(11\), the fourth number \(n\) satisfies \(41 + n = 60\), so \(n = 19\).

Equal lengths. Opposite sides of a rectangle or parallelogram are equal, and so are the equal sides of an isosceles triangle. If opposite sides are \((4x - 3)\) cm and \((2x + 9)\) cm, then \(4x - 3 = 2x + 9\), so \(x = 6\) and each side is \(21\) cm.

How to solve a linear equation

Clear fractions: multiply every term on both sides by the LCD.
Expand any brackets, taking care with negative signs.
Collect the pronumeral terms on one side and the numbers on the other.
Divide both sides by the coefficient of the pronumeral.
Check by substituting into the original equation.

How to solve a word problem

Define a pronumeral for the unknown quantity.
Write an equation from the facts, such as a total cost, a perimeter or a sum of ages.
Solve it, then answer the question in words, with units.

Example 1 — Brackets inside brackets

Solve \(2[3x - (x + 5)] = 3(x - 4)\).

Solution

Work from the inside out. Remove the round brackets first: the minus sign changes both signs inside them.

\(2[3x - x - 5]\)	\(=\)	\(3(x - 4)\)
\(2(2x - 5)\)	\(=\)	\(3x - 12\)
\(4x - 10\)	\(=\)	\(3x - 12\)
\(x\)	\(=\)	\(\boldsymbol{-2}\)

x = - 2

Check: left side \(2[-6 - 3] = -18\); right side \(3(-6) = -18\). Both sides agree.

Example 2 — One algebraic fraction

Solve \(\dfrac{2x + 7}{3} = x + 1\).

Solution

Multiply both sides by \(3\) to clear the fraction. Put brackets around \(x + 1\), because all of it is multiplied.

\(\dfrac{2x + 7}{3}\)	\(=\)	\(x + 1\)
\(2x + 7\)	\(=\)	\(3(x + 1)\)
\(2x + 7\)	\(=\)	\(3x + 3\)
\(7\)	\(=\)	\(x + 3\)
\(\therefore\ x\)	\(=\)	\(\boldsymbol{4}\)

\frac{2 x + 7}{3} = x + 1, x = 4

Check: \(\dfrac{8 + 7}{3} = 5\) and \(4 + 1 = 5\).

Example 3 — Two fractions

Solve \(\dfrac{x + 4}{3} - \dfrac{x - 2}{5} = 2\).

Solution

Multiply every term by the LCD, \(15\). Each fraction cancels to a whole number times its numerator, in brackets.

\(5(x + 4) - 3(x - 2)\)	\(=\)	\(30\)
\(5x + 20 - 3x + 6\)	\(=\)	\(30\)
\(2x + 26\)	\(=\)	\(30\)
\(2x\)	\(=\)	\(4\)
\(\therefore\ x\)	\(=\)	\(\boldsymbol{2}\)

5 (x + 4) - 3 (x - 2) = 30

The right side is multiplied by \(15\) too: \(2 \times 15 = 30\). Check: \(\dfrac{6}{3} - \dfrac{0}{5} = 2\).

Example 4 — An age problem

Kim is three times as old as her son Leo. In \(12\) years' time, Kim will be twice as old as Leo. How old are they now?

Solution

Let Leo's age now be \(l\) years, so Kim is \(3l\). In \(12\) years they will be \(l + 12\) and \(3l + 12\).

\(3l + 12\)	\(=\)	\(2(l + 12)\)
\(3l + 12\)	\(=\)	\(2l + 24\)
\(l\)	\(=\)	\(\mathbf{12}\)

l = 12

Leo is \(\mathbf{12}\) and Kim is \(\mathbf{36}\). Check: in \(12\) years they will be \(24\) and \(48\), and \(48 = 2 \times 24\) ✓

Common pitfalls

Multiply every term. When clearing fractions, the terms with no fraction are multiplied by the LCD too. In Example 3 the \(2\) on the right becomes \(30\).

Brackets after a minus sign. \(-3(x - 2)\) is \(-3x + 6\). Dropping the bracket gives \(-3x - 2\) or \(-3x - 6\), both wrong.

Change sides, change operation. Moving \(-x\) from the right means adding \(x\) to both sides, not subtracting it.

Answer the question asked. In a word problem, \(l = 12\) is not the full answer. Write each person's age, or the number of weeks, or the cost, as the question asks.

Frequently asked questions

How do you solve an equation with x on both sides?

Expand any brackets, then add or subtract an \(x\) term on both sides so the pronumeral is on one side only. Then solve as a two-step equation.

How do you solve an equation with fractions?

Multiply every term on both sides by the lowest common denominator. Keep each numerator in brackets as you do it. The fractions disappear and you solve the equation that is left.

How do you check the answer to an equation?

Substitute your answer into the original equation and work out each side separately. If both sides give the same number, the answer is correct.

Can the answer to an equation be a fraction or negative?

Yes. Many equations have a fraction, decimal or negative solution. For example, \(5x = 19\) gives \(x = \dfrac{19}{5} = 3.8\).

How do you turn a word problem into an equation?

Choose a pronumeral for the unknown, write the other quantities in terms of it, then use the fact in the question, such as a total or a perimeter, to write the two equal sides.

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Adding and Subtracting Algebraic Fractions

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Linear Inequalities

Solving Linear Equations

📖 Theory

How to solve a linear equation

How to solve a word problem

Common pitfalls

Frequently asked questions

Theory