MAT1793-ABED_1: Income - Getting Into It

Getting Into It

Equation Solving

When converting units, it is useful to be able to solve equations like 5x = 15 and \(\frac{t}{4}=\frac{5}{12}\).

Recall that solving an equation means finding a value for the variable that will make the equation true. A common method for solving equations is to apply operations to both sides of the equation to isolate the variable.

EXAMPLE 1

Solve 5x = 15 for x.

Dividing both sides of the equation by 5 will isolate x on the left side of the equation.

\(\begin{align} 5x&=15 \\ \\ \frac{5x}{\color{red}{5}}&=\frac{15}{\color{red}{5}} \\ \\ \frac{\cancel{5}x}{{\cancel{5}}}&=3 \\ \\ x&=3 \\ \end{align}\)

Verify the solution by substituting x = 3 into the original equation to make sure the left side is equal to the right side.

Left Side	Right Side
5x 5(3) 15	15
LS = RS

The left side equals the right side, so 3 is a solution to 5x = 15.

EXAMPLE 2

Solve \(\frac{t}{4}=\frac{5}{12}\).

Multiplying both sides of the equation by 4 will isolate t on the left side of the equation.

If you prefer to use a decimal, \(\frac{5}{12}\) × 4 can be evaluated on a calculator as \(5÷12×4=1.\overline{6}\) . The line on top of the 6 represents a \(1.\overline{6}=1.666666...\) . Both \(1.\overline{6}\) and 5/3 are acceptable answers.

\(\begin{align} \frac{t}{4}&=\frac{5}{12} \\ \\ \frac{t}{4}\times \color{red}{4}&=\frac{5}{12}\times \color{red}{4} \\ \\ \frac{t}{\cancel{4}}\times \cancel{4}&=\frac{5}{12}\times 4 \\ \\ t&=\color{red}{\frac{5\times 4}{12}} \\ \\ t&=\frac{20}{12} \\ \\ t&=\frac{5}{3} \\ \end{align}\)

Again, the solution can be verified by substituting \(t=1.\overline{6}\) into the original equation to see if the left side is equal to the right side.

Left Side	Right Side
\(\frac{t}{4}\) \(\frac{\color{red}{1.\overline{6}}}{4}\) \(0.41\overline{6}\)	\(\frac{5}{12}\) \(0.41\overline{6}\)
LS = RS