Solving linear equations – Linear Functions, Equations and Inequalities (Senior 1 Mathematics) | Ganzaa

Linear Functions, Equations and Inequalities

Lessons

Senior 1 · Mathematics

Linear Functions, Equations and Inequalities

5 lessons

Lesson 4 of 5

Solving linear equations

25 min

A shopkeeper puts two identical bags of sugar and three 1 kg weights on one pan of a balance, and eleven 1 kg weights on the other. The beam is level. How heavy is one bag? Take 3 kg off each pan and the beam stays level: two bags balance 8 kg. Halve both sides: one bag is 4 kg. Solving an equation in algebra follows exactly these moves.

Definition

Equation: a statement that two expressions are equal, written with an equals sign. Example: $2 x + 3 = 11$ .

Linear equation in one unknown: an equation in which the unknown appears only to the power 1. It can always be rearranged to $a x + b = 0$ with $a \ne 0$ . Example: $5 x - 7 = 2 x + 8$ is linear; $x^{2} = 9$ is not.

Definition

Solution (or root): a value of the unknown that makes both sides equal. To solve an equation is to find its solution.

Example: $x = 4$ is the solution of $2 x + 3 = 11$ , because $2 (4) + 3 = 11$ .

The rule: keep the balance level

You may do any of these to an equation, as long as you do it to both sides:

add the same number, or subtract the same number;
multiply by the same number, or divide by the same number, provided it is not zero.

Why it works. The two sides are the same number written in two ways. Doing the same thing to equal numbers gives equal results, just as taking 3 kg off both pans keeps the beam level. Zero is excluded because multiplying both sides by 0 turns every equation into $0 = 0$ , which tells you nothing, and division by 0 has no meaning.

Important

A strategy for any linear equation

Clear fractions: multiply every term on both sides by the lowest common multiple (LCM) of the denominators.
Remove brackets.
Collect the unknown terms on one side and the numbers on the other.
Divide both sides by the coefficient of the unknown.
Check: substitute your answer into the original equation.

Worked example

Solve $2 x + 3 = 11$

\begin{aligned} 2x + 3 &= 11 \\ 2x + 3 - 3 &= 11 - 3 && \text{(subtract 3 from both sides)} \\ 2x &= 8 \\ \frac{2x}{2} &= \frac{8}{2} && \text{(divide both sides by 2)} \\ x &= 4 \end{aligned}

Check: $2 (4) + 3 = 11$ . ✓

Worked example

Solve $5 x - 7 = 2 x + 8$

\begin{aligned} 5x - 2x - 7 &= 8 && \text{(subtract } 2x \text{ from both sides)} \\ 3x &= 15 && \text{(add 7 to both sides)} \\ x &= 5 && \text{(divide both sides by 3)} \end{aligned}

Check: left side $5 (5) - 7 = 18$ ; right side $2 (5) + 8 = 18$ . ✓

Worked example

Solve $3 (x - 2) = 2 (x + 1) + 4$ . (Brackets.)

\begin{aligned} 3x - 6 &= 2x + 2 + 4 && \text{(remove brackets)} \\ 3x - 6 &= 2x + 6 \\ x &= 12 && \text{(subtract } 2x \text{, add 6)} \end{aligned}

Check: $3 (10) = 30$ and $2 (13) + 4 = 30$ . ✓

Worked example

Solve (a) $\frac{x}{3} + \frac{x}{4} = 7$ ; (b) $\frac{x + 1}{2} - \frac{x - 2}{5} = 3$ . (Fractions.)

(a) The LCM of 3 and 4 is 12. Multiply every term by 12: $4x + 3x = 84 \;\Rightarrow\; 7x = 84 \;\Rightarrow\; x = 12.$ Check: $\frac{12}{3} + \frac{12}{4} = 4 + 3 = 7$ . ✓

(b) The LCM of 2 and 5 is 10. Multiply every term by 10, keeping each numerator in brackets: $\begin{aligned} 5(x + 1) - 2(x - 2) &= 30 \\ 5x + 5 - 2x + 4 &= 30 \\ 3x + 9 &= 30 \\ x &= 7 \end{aligned}$ Check: $\frac{8}{2} - \frac{5}{5} = 4 - 1 = 3$ . ✓

Solving problems with equations

Choose a letter for the unknown and say what it stands for, with units.
Write an equation from the facts.
Solve it.
Answer the question in words, with units.
Check the answer against the story, not only the equation.

Worked example

Keza pays 7,700 Frw for 3 kg of rice and 2 kg of sugar. Sugar costs 1,600 Frw per kg. What is the price of 1 kg of rice?

Let $r$ be the price of 1 kg of rice, in Frw.

\begin{aligned} 3r + 2(1{,}600) &= 7{,}700 \\ 3r + 3{,}200 &= 7{,}700 \\ 3r &= 4{,}500 \\ r &= 1{,}500 \end{aligned}

Rice costs 1,500 Frw per kg. Check: $3 \times 1{,}500 + 2 \times 1{,}600 = 4{,}500 + 3{,}200 = 7{,}700$ . ✓

Worked example

Three friends buy a football for 13,000 Frw. Mugisha pays twice as much as Habimana, and Ishimwe pays 1,000 Frw more than Habimana. How much does each pay?

Let $h$ be Habimana's share in Frw. Then Mugisha pays $2 h$ and Ishimwe pays $h + 1,000$ .

h + 2h + (h + 1{,}000) = 13{,}000 \;\Rightarrow\; 4h = 12{,}000 \;\Rightarrow\; h = 3{,}000.

Habimana pays 3,000 Frw, Mugisha 6,000 Frw and Ishimwe 4,000 Frw. Check: $3,000 + 6,000 + 4,000 = 13,000$ . ✓

Warning

Multiply every term by the LCM. A common error in $\frac{x}{3} + \frac{x}{4} = 7$ is to write $4 x + 3 x = 7$ , forgetting to multiply the 7 by 12. The number on the right is a term too.

Warning

A minus sign in front of a fraction applies to the whole numerator. In $-\frac{x - 2}{5}$ , multiplying by 10 gives $- 2 (x - 2) = - 2 x + 4$ , not $- 2 x - 4$ . Keep the numerator in brackets until the bracket is removed.

Tip

"Move a term to the other side and change its sign" is only a shortcut for "add or subtract it on both sides". If you are ever unsure, write the full step: it is harder to make a sign mistake that way. And always check: a correct root makes both sides equal.

Math Antics (mathantics): "Algebra Basics: Solving 2-Step Equations". Watch why the adding or subtracting step comes before the dividing step.

✦

Practice

Solve each equation and check your answer.

$x + 9 = 4$
$4 x - 7 = 21$
$7 - 3 x = 19$
$6 x + 5 = 2 x + 17$
$4 (x + 3) = 2 (x - 1) + 24$
$\frac{x}{2} + \frac{x}{5} = 14$
$\frac{2x - 1}{3} = \frac{x + 4}{2}$
The perimeter of a rectangular garden is 46 m, and its length is 5 m more than its width. Find its length and width.

Show answersHide answers

$x = - 5$
$4 x = 28$ , so $x = 7$
$- 3 x = 12$ , so $x = - 4$
$4 x = 12$ , so $x = 3$
$4 x + 12 = 2 x + 22$ , so $2 x = 10$ and $x = 5$
Multiply by 10: $5 x + 2 x = 140$ , so $x = 20$
Multiply by 6: $2 (2 x - 1) = 3 (x + 4)$ , so $4 x - 2 = 3 x + 12$ and $x = 14$ . Check: $\frac{27}{3} = 9 = \frac{18}{2}$ .
Width $w$ : $2 (w + w + 5) = 46$ , so $4 w + 10 = 46$ and $w = 9$ . The garden is 14 m long and 9 m wide.

Sources and credits

2.4: Use a General Strategy to Solve Linear Equations, from Elementary Algebra by Lynn Marecek, MaryAnne Anthony-Smith and Andrea Honeycutt Mathis, OpenStax, provided by LibreTexts, licensed under CC BY 4.0. Adapted by Ganzaa.
2.5: Solve Equations with Fractions or Decimals, from Elementary Algebra by Lynn Marecek, MaryAnne Anthony-Smith and Andrea Honeycutt Mathis, OpenStax, provided by LibreTexts, licensed under CC BY 4.0. Adapted by Ganzaa.
Video: Algebra Basics: Solving 2-Step Equations, mathantics, embedded from YouTube.

The worked examples, diagrams, practice items and quiz questions were written by Ganzaa.

1The rule: keep the balance level

2Solving problems with equations

The rule: keep the balance level

Solving problems with equations