Simplifying and Solving Equations

You have learned how to solve equations of the form $ax+b=cx+d$ where $a$, $b$, $c$, and $d$ are numbers. You have also learned how to simplify various complicated expressions into the form $ax+b$. Putting these techniques together, in this lesson you will solve various complicated equations, by first simplifying each side of each equation. You will also learn that not all equations have exactly one solution.

Simplifying equations before solving them

We’ll begin by looking at the equation

$$ 2(x+3)=8 $$

which is drawn on the grids to the left.

Find the solution to this equation by sliding the $x$ slider below the grids.

This equation is not in the form $ax+b=cx+d$, but we can convert it to this form by using the distributive law to expand the left-hand side.

Using the distributive law, expand the expression $2(x+3)$ as shown in the table below.

If you replace $2(x+3)$ by its expanded version in the equation $2(x+3)=8$, what do you get?

In simp-eq-qn, you found that the equation $2(x+3)=8$ could be simplified into $2x+6=8$. Solve this equation by adding the same number to both sides. Then check your solution.

Is this the same solution you found in simp-eq-qn?

Simplify each equation in the table below into the form $ax+b=cx+d$. Then solve that equation and check your solution, using scratch paper.

Equations without a unique solution

Now we’ll look at the equation

$$ 2(2x-1)=4x-1 $$

which is illustrated on the grids to the left.

Can you find any value of $x$ which is a solution to this equation by sliding the sliders?

Simplify this equation into the form $ax+b=cx+d$.

When you want to solve an equation of the form $ax+b=cx+d$, you usually start by subtracting $cx$ from both sides. What do you get when you do that to this equation?

As you can see, this equation can be simplified into one that is never true (no matter what $x$ is). That means there is no $x$ value which solves the equation, so we say that the equation has no solutions.

Let’s try to solve the equation

$$ 2(x+3)=2x+6 $$

which is illustrated on the grids to the left.

Look at the grids to the left. Is $0$ a solution to this equation?
Slide the slider to $x=1$. Is $1$ a solution to this equation?
By sliding the slider, set $x$ to any other number that you choose. Is that number a solution to this equation?

Simplify the equation $2(x+3)=2x+6$ into the form $ax+b=cx+d$.

What do you get when you subtract $cx$ from both sides of this equation?

As you can see, this equation can be simplified into one that is always true (no matter what $x$ is). That means there are an infinite number of solutions!

In general:

If an equation can be simplified into the form $p=q$, where $p$ and $q$ are different numbers, it has no solutions.

If an equation can be simplified into the form $x=p$, where $p$ is some number, it has one solution.

If an equation can be simplified into the form $p=p$, it is always true, and has an infinite number of solutions.

Write each equation in the table below in the form $ax+b=cx+d$. Then determine if the number of solutions to that equation is zero, one, or infinite, using scratch paper if necessary.