3. Solve the following system of equations by

- a) Graphing
- b) Substitution
- c) Addition

and check your answer

Graph both equations. Solve the equations for *y.* In the first one

2*x* + 3*y* = 3

transpose the *x* term

3*y* = -2*x* + 3

and divide both sides by 3.

Do the same thing for the other equation

3*x* + 2*y* = 7

Transpose

2*y* = -3*x* + 7

and divide by 2.

Now make up tables of *x*'s and *y*'s for both equations.

Plot the points and draw the graphs. We see that the point of intersection is (3, -1), so the solution is

*x* = 3

and

*y* = -1

If the point where the lines intersect has integer coordinates, the graphing method works quite well. But when the point does not have integer coefficients, it is often hard to tell fractional coordinates by looking at the graph.

If we substitute these numbers into the original equations, in the first equation we get

2(3) + 3(-1) = 3

or

6 - 3 = 3

and in the second equation we get

3(3) + 2(-1) = 7

or

9 - 2 = 7

and the numbers check in both equations.

b. Substitution

Solve one of the equations for one of the unknowns. In this case it is easiest to solve the first equation for *y*.

Now substitute this solution into the other equation.

At this point we could combine the *x* terms. To subtract the coefficients, 3 - 4/3 = 5/3, but it might be easier to first clear denominators by multiplying both sides of the equation by 3.

9*x* - 4*x* + 6 = 21

5*x* + 6 = 21

Subtract 6 from both sides.

5*x* = 15

Divide both sides by 5.

*x* = 3

Remove parentheses

*x* = 3

To find *y*, substitute this solution into the equation where we solved for *y* as a function of *x*.

*y* = -2 + 1

*y *= -1

which is the same answer we got by graphing.

Multiply the equations by suitable numbers so that the coefficients on one of the unknowns match up. In this case, if we multiply the first equation by 3 and the second equation by -2,
the coefficients of the *x*'s will match up and they will go away when
we add.

*y* = -1

To find *x*, we substitute this value of *y* into either of the original equations. The first equation might be easiest.

2*x* + 3(-1) = 3

2*x *- 3 = 3

Add 3 to both sides.

2*x* = 6

Divide both sides by 2.

*x* = 3

and we get the same answer we got the other two ways.