Linear Equations
Learning objectives
- Solve one-step and multi-step linear equations and check the answer.
- Solve equations with the unknown on both sides, with brackets and with fractions.
- Recognise equations with no solution or infinitely many solutions.
- Turn word problems into equations and connect solutions to graphs.
Equations and solutions
An equation states that two expressions are equal. A solution is a value of the unknown that makes it true. A linear equation in one unknown has the form \(ax+b=c\) with \(a\neq 0\).
Think of an equation as a balanced scale. Whatever you do to one side you must also do to the other: add or subtract the same number, or multiply or divide by the same nonzero number.
If \(a\neq 0\), the equation \(ax+b=c\) has exactly one solution: \(x=\frac{c-b}{a}\).
Show proof
Subtracting \(b\) from both sides gives \(ax=c-b\). Dividing both sides by \(a\neq 0\) gives \(x=\frac{c-b}{a}\). Each step can be reversed, so this value is a solution and the only one. ∎
Try it and see the graph
Change \(a\), \(b\) and \(c\). The steps are shown for \(ax+b=c\). In the graph, the blue line is \(y=ax+b\), the dashed line is \(y=c\), and the green point where they meet has the solution as its \(x\)-coordinate.
Worked examples
Solve \(3x-7=11\).
\[3x=18,\qquad x=6\]
Check: \(3\cdot 6-7=11\).
Solve \(5x+2=3x+10\).
Subtract \(3x\) and 2 from both sides: \(2x=8\), so \(x=4\). Check: \(5\cdot 4+2=22=3\cdot 4+10\).
Solve \(2(x+3)=14\).
Divide by 2: \(x+3=7\), so \(x=4\). (Or expand: \(2x+6=14\).)
Solve \(\frac{x}{3}+2=5\).
Subtract 2: \(\frac{x}{3}=3\). Multiply by 3: \(x=9\).
Some equations behave differently. \(2x+1=2x+5\) leads to \(1=5\), which is false for every \(x\): no solution. \(3(x+1)=3x+3\) leads to \(3=3\), which is true for every \(x\): infinitely many solutions.
Set \(a=0\) in the solver above to see both cases.
Word problems
A number is tripled and then increased by 5, giving 26. Find the number.
Let the number be \(x\). Then \(3x+5=26\), so \(3x=21\) and \(x=7\).
Ali is 4 years older than Sara and the sum of their ages is 28. Find their ages.
Let Sara be \(s\) years old. Then \(s+(s+4)=28\), so \(2s=24\), \(s=12\), and Ali is 16.
Exercises
Solve \(x+9=17\).
Show solution
\(x=17-9=8\)
Solve \(4x-3=21\).
Show solution
Add 3 to both sides: \(4x=24\), so \(x=6\).
Solve \(7x-5=4x+16\).
Show solution
Subtract \(4x\) and add 5: \(3x=21\), so \(x=7\). Check: \(44=44\).
Solve \(3(2x-1)=15\).
Show solution
Divide by 3: \(2x-1=5\), so \(2x=6\) and \(x=3\).
Solve \(\frac{x+1}{2}=5\).
Show solution
Multiply by 2: \(x+1=10\), so \(x=9\).
A rectangle has perimeter 38 and its length is 3 more than its width. Find its sides.
Show solution
Let the width be \(w\). Then \(2(w+w+3)=38\), so \(4w+6=38\), \(w=8\), and the length is 11.
How many solutions does \(4x+2=4x+7\) have?
Show solution
Subtracting \(4x\) gives \(2=7\), which is false, so there is no solution.
Summary
- Do the same operation to both sides to keep an equation balanced.
- The equation \(ax+b=c\) with \(a\neq 0\) has the single solution \(x=\frac{c-b}{a}\).
- Always check a solution by substituting it back.
- Some equations have no solution, and some are true for every \(x\).