Solving Equations Graphically

Solving $ax+b=c$ graphically

One way to solve an equation is to graph both the left-hand side and the right-hand side, and see where they are equal.

We will start by looking at the equation $x+2=4$. The graphs of $y_L=x+2$ (the left-hand side of the equation) and $y_R=4$ (the right-hand side) are shown to the left. Notice that the two graphs intersect (meet) at the point $(2,4)$.

What is the value of $x$ at the point where the two graphs intersect?

Check to see if this value of $x$ solves the equation $x+2=4$.

If an $x$-value makes the two sides of an equation equal, then the graphs of the two sides intersect at that $x$-value. So:

If you graph the two sides of an equation, a solution to that equation is an $x$-value where the two graphs intersect.

We will now look at the equation $x-2=1$. Notice that the graphs of $y_L=x-2$ and $y_R=1$ are now shown to the left.

What is the intersection point of the two graphs?

What is the value of $x$ at that point?

What value of $x$ solves the equation $x-2=1$?

Check that this value of $x$ actually does solve the equation $x-2=1$.

The two lines graphed on the left give you a picture of the equation $x-2=1$. By seeing where the lines intersect, you can use that picture to solve the equation.

For each row of the table, use the graphs to the left to solve the given equation. Check your solution.

Equation	Intersection and solution	Checking

The two sides of the equation $3x-1=3$ are graphed on the grid to the left.

Can you tell exactly where the red and blue lines intersect by looking at the graph?

Can you tell what the solution to the equation $3x-1=3$ is by looking at the graph?

Even when the graph doesn’t show you the exact solution to the equation, it can still tell you something about the value of that solution.

Which two integers are closest to the solution to the equation? (An integer is a whole number like 0, 1, 2, ..., or the negative of a whole number.)

and

Solving $ax+b=cx+d$ graphically

The graphs of $y_L=2x$ and $y_R=-x+3$ are shown to the left.

What is the intersection point of the two graphs?

What value of $x$ solves the equation $2x=-x+3$?

Check your solution.

For each row of the table, use the graphs on the left to solve the given equation. Check your solution.

Equation	Intersection and solution	Checking

Click . The two sides of the equation $$1/4x+2=3/4x+1$$ are now graphed on the grid to the left.

What is the $x$-value at their point of intersection?

Check that this $x$-value solves the equation $$1/4x+2=3/4x+1$$. (Remember to use improper fractions.)

The two sides of the equation $5x-4=2x+1$ are graphed on the grid to the left.

Can you tell what the solution to the equation $5x-4=2x+1$ is by looking at the graph?

Which two integers are closest to the solution to the equation?

and

Click . Now the two sides of the equation $2x+3=-4x+1$ are graphed on the grid to the left.

Which two integers are closest to the solution to this equation?

and

Picturing algebraic steps to solve equations

The two sides of the equation $3x-1=3$ are graphed on the grid to the left. If we wanted to solve this equation algebraically, we would start by adding the same number to both sides of the equation, in order to isolate the term with an $x$ in it.

The $k$ slider below the grid adds $k$ to both sides of the equation, allowing you to see the equation $3x-1+k=3+k$ on the grid to the left. As $k$ changes, does the $x$-coordinate of the intersection point change?

Does the $y$-coordinate of the intersection point change?

Does the solution to the equation change?

Slide $k$ to $1$. Do the equations under the grid get simpler?

Solve the equation $3x-1=3$ algebraically, and check your solution.

Now the two sides of the equation $5x-4=2x+1$ are graphed on the grid to the left. To solve this equation algebraically, we would start by adding the same multiple of $x$ to both sides, to make it so that $x$ appears on only one side of the equation.

In addition to the $k$ slider, there is now a $j$ slider below the grid, which adds $jx$ to both sides of the equation. As $j$ changes, does the $x$-coordinate of the intersection point change?

Does the $y$-coordinate of the intersection point change?

Does the solution to the equation change?

Slide $j$ to $-2$. Do the equations under the grid get any simpler?

What value would you have to slide $k$ to in order to make the equations under the grid simpler?

Solve the equation $5x-4=2x+1$ algebraically, and check your solution.