# Solving \$ax+b=c\$

## Multiplying both sides of an equation by a number

The equation \$3x=6\$ is illustrated on the two grids to the left, with a slider for \$x\$. There is also an \$m\$ slider, which changes the equation by multiplying \$3x\$ and \$6\$ by the same number \$m\$.

For each row of the table below, slide the \$m\$ slider to the value given. Then type in the equation that is illustrated by the two grids. Finally, use the \$x\$ slider to find the value of \$x\$ that solves that equation.

After sliding the \$m\$ slider, you can simplify the equation you get. For example, the equation \$2x=6\$ is currently pictured on the grids to the left. So if you slide the \$m\$ slider to \$m=2\$, you will get a picture of the equation \$2(2x)=2(6)\$, which can be simplified to \$4x=12\$.

The equation \$2x=6\$ is pictured on the grids to the left. Change this equation by sliding the \$m\$ slider to each of the values in the table below. Then simplify and solve the resulting equation.

Changing the \$m\$ slider doesn’t change the solution to an equation. So you can slide \$m\$ to some value that makes the equation simpler.

Each row of the table below has an equation. Find the \$m\$ that isolates \$x\$ on one side of that equation. Then find and check the solution to the equation.

 Look at the last problem in the table. What value of \$m\$ isolates \$x\$ in the equation \$4x=6\$?
 Turn that \$m\$ into a fraction.

Notice that, to solve an equation where \$x\$ was multiplied by 4, you multiplied everything by \$\$1/4\$\$. In general:

The equation \$ax=c\$ can be made simpler by multiplying both sides of it by \$\$1/a\$\$.

Solve each equation in this table by multiplying both sides by the appropriate fraction, and then simplifying. (Write fractional answers in lowest terms. If a fractional answer is negative, put the minus sign in the numerator.)

The two grids to the left show the equation \$4x=12\$, with an extra \$d\$ slider underneath both grids. This slider changes the equation by dividing both sides by the same number.

For each row of the table below, slide the \$d\$ slider to the value given, and type in the equation that is now illustrated on the two grids. Then, find the value of \$x\$ that solves the equation.

 What is a simpler way of writing \$\${4x}/4\$\$?

That is, dividing by \$d\$ works just like multiplying by \$\$1/d\$\$: it doesn’t change the answer, and it can be used to make the equation simpler.

The equation \$ax=c\$ can be made simpler by dividing both sides of it by \$a\$.

Solve these equations of the form \$ax=c\$ by multiplying both sides by \$\$1/a\$\$, and check your answer. (If \$\$a=p∕q\$\$ is a fraction, then \$\$1/a = 1 / {p∕q} = q/p\$\$.)

## Solving \$ax+b=c\$

The equation \$3x-2=7\$ is pictured on the grid, along with a \$k\$ slider that adds the same number to both sides. If you want to solve this equation, you can start by adding \$2\$ to both sides and simplifying. This turns the equation into \$3x=9\$, which the last few questions have taught you how to solve.

Each row of the table below has an equation in it. By adding the same number to both sides, simplify that equation into one that looks like “\$ax\$ = something” for some \$a\$.

Each row of the table below has an equation in it. Simplify that equation into one that looks like “\$ax\$ = something” for some \$a\$, and then solve it. Then check your answer.