5.1Linear Equations: Concept and Solving

An equation says two expressions are equal. To solve it is to find the value of the unknown that makes that true.

Do the same thing to both sides. An equation is a balance: change one side alone and it stops balancing.

Balancing principle

\displaystyle a = b \;\Rightarrow\; a \pm c = b \pm c, \quad ac = bc, \quad \frac{a}{c} = \frac{b}{c} \; (c \neq 0)

Both pans change by the same amount, so the beam stays level. That is why the equation still holds.

Add or subtract the same number on both sides to move a constant.
Multiply or divide both sides by the same non-zero number to move a coefficient.
Expand any brackets first, then collect the unknown on one side.

Two-step equation

Solve $\displaystyle 5x - 7 = 18$ .

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\displaystyle 5x - 7

\displaystyle = 18

\displaystyle 5x

\displaystyle = 25 \quad \scriptsize\textit{add 7 to both sides}

\displaystyle x

\displaystyle = 5 \quad \scriptsize\textit{divide both sides by 5}

Now you try

Solve $4a - 2 = 34$ .

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\displaystyle 4a - 2

\displaystyle = 34

\displaystyle 4a

\displaystyle = 34 + 2 \quad \scriptsize\textit{add 2 to both sides}

\displaystyle 4a

\displaystyle = 36

\displaystyle a

\displaystyle = \frac{36}{4}

\displaystyle a

\displaystyle = 9

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Solve $\displaystyle 7y - 3 = 4y + 15$ .

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\displaystyle 7y - 3

\displaystyle = 4y + 15

\displaystyle 3y - 3

\displaystyle = 15 \quad \scriptsize\textit{subtract 4y from both sides}

\displaystyle 3y

\displaystyle = 18 \quad \scriptsize\textit{add 3 to both sides}

\displaystyle y

\displaystyle = 6

Solve $\displaystyle 3(2m + 1) = 4(m - 5)$ .

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\displaystyle 6m + 3

\displaystyle = 4m - 20 \quad \scriptsize\textit{expand both sides}

\displaystyle 2m

\displaystyle = -23 \quad \scriptsize\textit{collect terms}

\displaystyle m

\displaystyle = -\frac{23}{2}

Now you try

Solve $8x - 4 = 5x + 20$ .

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\displaystyle 8x - 4

\displaystyle = 5x + 20

\displaystyle 8x - 5x

\displaystyle = 20 + 4 \quad \scriptsize\textit{collect like terms}

\displaystyle 3x

\displaystyle = 24

\displaystyle x

\displaystyle = \frac{24}{3}

\displaystyle x

\displaystyle = 8

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The solution to $\displaystyle ax - 3 = 2x + 9$ is $\displaystyle x = 4$ . Find $\displaystyle a$ .

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Substitute

x = 4

into the equation:

\displaystyle a(4) - 3

\displaystyle = 2(4) + 9

\displaystyle 4a - 3

\displaystyle = 8 + 9

\displaystyle 4a - 3

\displaystyle = 17

\displaystyle 4a

\displaystyle = 20

\displaystyle a

\displaystyle = 5

Check:

5(4) - 3 = 17

and

2(4) + 9 = 17

✓