Solving $ax+b=cx+d$

Review: Solving $ax+b=d$

Solve each of these equations, and check your solutions.

Equations with $x$ on both sides

The equation $5x=2x+6$ is illustrated on the two grids to the left, with a slider for $x$. Because this equation has an $x$ on both sides, sliding the slider changes both grids.

What $x$ makes both grids have the same value (number of green squares minus number of pink squares)?

Check your answer.

The table below has several equations with $x$ on both sides, which are pictured on the grids to the left. Solve each equation by sliding the slider to find the $x$ which makes both grids have the same value.

Adding $jx$ to both sides

As in bothq1, the equation $5x=2x+6$ is illustrated on the two grids to the left, with a slider for $x$. There is now also a $j$ slider, which changes the equation by adding the expression $jx$ to both sides.

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

Notice that sliding the $j$ slider doesn’t affect which $x$ value solves the equation.

After sliding the $j$ slider, you can simplify the equation you get by combining like terms. For example, the equation $4x+1=2x+7$ is currently pictured on the grids to the left. If you slide the $j$ slider to $j=3$, you will get a picture of the equation $3x+4x+1=3x+2x+7$, which can be simplified to $7x+1=5x+7$.

The equation $4x+1=2x+7$ is pictured on the grids to the left. Change this equation by sliding the $j$ slider to each of the values in the table below. Then simplify the resulting equation, and slide the $x$ slider to find its solution.

Because changing the $j$ slider and simplifying doesn’t change the solution to an equation, you can slide $j$ to some value that makes the whole equation simpler. You should be able to find a $j$ that puts all the $x$ terms on the left side of the equals sign. Then your equation will be like the ones in reviewQn, which you already know how to solve.

Each row of the table below has an equation to solve. Find the $j$ that eliminates the $x$ term on the right side of the equals sign in that equation.

Notice that, to simplify the equation $5x+6 = 3x+2$, we added $-3x$ to both sides. In general:

The equation $ax+b=cx+d$ can be made simpler by adding $-cx$ to both sides.

Simplify each equation by adding an appropriate multiple of $x$ to both sides.

Solving $ax+b=cx+d$

In the next few questions, we will carefully go through the steps to solve the equation $4x+1=2x+7$.

The grids to the left show the equation $4x+1=2x+7$. Slide the $j$ slider to simplify this equation, and type in your result below.

You have simplified the equation to $2x+1=7$, which is now shown on the grids to the left. Slide the $k$ slider to further simplify this equation, and type in your result below.

Now you’ve further simplified the equation to $2x=6$, and the grids to the left show this equation. Slide the $d$ slider to perform a final simplification. Solve the equation and type in your result below.

Check that the $x$ you just found solves the original equation.

Solve these equations, using the grids to the left if necessary, and check your solutions.