# Solving \$x+b=c\$

## Equations on two grids

 Look at the grids to the left. What \$x\$ makes them both have the same value (number of green squares minus number of pink squares)?

Notice that by answering this question, you found a solution to the equation \$x-3=1\$. That is, the two grids to the left illustrate this equation.

Whenever you solve an equation, it’s a good idea to check your solution by substituting it back into the original equation. Then make sure that both sides of the result are equal.

 Check your solution to this equation.

Solve the equation in each row of the table below by finding the value of \$x\$ that makes both grids to the left have the same value.

## Adding to both sides of an equation

Look at the two grids to the left, which currently illustrate the equation \$x-5=2\$. A second slider has been added below the grids. This slider changes the equation by adding the same number \$k\$ to both \$x-5\$ and \$2\$.

For each row of the table below, slide the \$k\$ slider to the value given in the table. 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, as in the previous question.

If an \$x\$ value makes both sides of an equation equal, then the sides stay equal for that \$x\$ after you add some \$k\$ value to both sides.

After sliding the \$k\$ slider, you can simplify the equation you get. For example, the equation \$x+2=6\$ is currently pictured on the grids to the left. If you slide the \$k\$ slider to \$k=1\$, you will get a picture of the equation \$x+2+1=6+1\$. But \$x+2+1\$ means the same thing as \$x+3\$, and \$6+1\$ means the same thing as \$7\$. So a simpler way to write \$x+2+1=6+1\$ would be \$x+3=7\$.

The equation \$x+2=6\$ is pictured on the grids to the left. Change this equation by sliding the \$k\$ slider to each of the values in the table below, simplify the resulting equation, and find the \$x\$ that solves it.

 Does sliding the \$k\$ slider (that is, adding the same number to both sides of the equation) change the value of \$x\$ that solves the equation? (Type yes or no.)

Notice that in the last row of the table, \$x\$ is alone on one side of the equation after simplifying. That is, we have isolated \$x\$ from the other terms in the equation.

Sliding the \$k\$ slider doesn’t change the solution to an equation. So you can slide \$k\$ to some value that makes the equation simpler. If you slide \$k\$ to a value which isolates \$x\$ — that is, puts \$x\$ alone on one side of the equation — you will have solved the equation.

For example, the grid on the left shows the equation \$x-1=4\$. If you set \$k=1\$, the value of the green square from the \$k\$ slider will cancel out the value of the pink square from the original equation, so the top grid will just have value \$x\$. That is, setting \$k=1\$ allows you to isolate \$x\$.

This means that you can solve the equation by adding 1 to both sides:

Click to see the equation \$x-2=5\$ pictured on the grids to the left. Then, solve that equation by finding the \$k\$ value that isolates \$x\$.

Whenever you find the solution to an equation in this way, you should check that it actually solves the equation.

Each row of the table below has an equation. Find the \$k\$ that isolates \$x\$ in the equation. Then find and check the solution to the equation.

Notice that you always want \$k\$ to be the opposite of whatever number \$x\$ is being added to. That is:

The equation \$x+b=c\$ can be made simpler by adding \$-b\$ to both sides of it (or subtracting \$b\$ from both sides of it).

If you want to solve an equation and you aren’t given this kind of grid picture, you can do the same thing. If the equation looks like \$x+b=c\$, you can solve it by:

• Adding \$-b\$ to both sides of the equation (or subtracting \$b\$ from both sides of the equation).
• Simplifying the two sides of the equation until \$x\$ is isolated.

Solve each equation in this table by adding the appropriate number to both sides and then simplifying. Then check your solution.