TUTORIAL

In this tutorial, you will learn how to solve a system of two equations using the substitution method.

Let’s start off with an example problem.

 

EXAMPLE: Solve for 3x+4y=-4 (We’ll call this EQUATION ONE)

                        And for x+y=2 (We’ll call this EQUATION TWO)

Now, let’s start off and solve the problem.

 

Step 1: Pick one equation to solve for x. A tip would be to solve for the equation that is easier to solve for (ie. An equation that already has x or without a coefficient or an easy-to-solve coefficient). In this case, let’s solve for EQUATION TWO

x+y=2

            x=-y+2 ←This is the equation solved for x

 

Step 2: Now that you’re done with that, substitute the value of x (the other side of “x=”) in the step above (EQUATION TWO) into EQUATION ONE.

            3x+4y=-4

            3(-y+2)+4y=-4

 

Step 3: After that, solve the above equation for y

3(-y+2)+4y=-4           

-3y+6+4y=-4

y+6=-4

y=-4-6

y=-10

 

Step 4: Now that you know the value of y, substitute it into EQUATION TWO’s “solved for x” form and find out the value of x.

            x=-(-10) +2

            x=10+2

            x=12

Since the y value is -10 and the x value is 12,  it would make the solution to the system (12,-10).

 

Step 5: You’ve solved the problem! Congratulations, but there’s still one more step… Check your answer! Substitute the values for x and y back into the original equations in order to do this.

EQUATION ONE:

3(12)+4(-10)=-4

36-40=-4

-4=-4 ☺

EQUATION TWO:

(12)+(-10)=2

2=2 ☺

                        Congratulations! Your Math Checks Out!

 

Overall, we’ve summed all this complicated stuff up into three steps… here they are:

1.)    Solve the easier-to-solve equation for one of its variables.

2.)    Substitute the aforesaid expression from the previous step into the other equation in the system, since there is only one unique variable left after this, solve for the other variable.

3.)    Substitute the solved variable’s value from step two into the changed equation (solved for x) in step one and solve. After that, check your answer, and you’re basically finished!

 

 

PRACTICE PROBLEMS:

Practice makes perfect! Solve these systems for practice.

 

1.)    3x+2y=10

2x-y=9

Ans: (4,-1)

 

2.)    -x+2y=3

4x-5y=-3

Ans: (3,3)

 

3.)    5x+6y=-45

x-0.5y=8

Ans: (3,-10)

 

4.)    4x+6y=15

-x+2y=5

Ans: (0, 2.5)

 

5.)    x+y=1

2x+y=2

Ans: (1,0)

 

6.)    2x+y=1

x+y=4

Ans: (-3,7)