THE SUBSITUTION METHOD

STEP 1 SOLVE ONE OF THE EQUATIONS FOR ONE OF ITS VARIABLES.

STEP 2 SUBSTITUTE THE EXPRESSION FROM STEP 1 INTO THE OTHER EQUATION AND SOVLE FOR THE OTHER VARIABLE.

STEP 3 SUBSTITUTE THE VALUE FROM STEP 2 INTO THE ORIGINAL EQUATION AND SOLVE.

 

EQUATION ONE: 3X + 4Y = -4
EQUATION TWO: X + 2Y = 2

1 SOLVE EQUATION TWO FOR X.
EQUATION TWO: X + 2Y = 2
REVISED EQUATION: X = -2Y + 2

2 SUBSTITUTE THE EXPRESSION FOR X INTO THE EQUATION ONE AND SOLVE FOR Y.

EQUATION ONE: 3X + 4Y = -4
SUBSTITUTE -2Y + 2 FOR X
EQUATION ONE: 3(-2Y+2) + 4Y = -4
SOLVE
Y= 5

3 SUBSTITUTE THE VALUE OF Y INTO THE ORIGINAL EQUATION TWO AND SOLVE FOR X.

EQUATION TWO: X + 2Y = 2
SUBSTITUTE 5 FOR Y
EQUATION TWO: X + 2(5) = 2
SOLVE
X=-8

THE sOLUTION IS (-8,5).

WHAT METHOD TO CHOOSE?

IN STEP ONE OF EXAMPLE ONE, YOU COULD HAVE SOLVED FOR X OR Y. iT IS EASIER TO SOLVE FOR X IN EQUATION TWO BECAUSE THE X-COEFFICIENT IS 1. yOU ALWAYS WANT TO SOLVE AN EQUATION FOR THE VARIABLE WHOSE COEFFICIENT IS 1 OR -1.

X - 5Y = 11 2X + 7Y = -3 Solve for x

8X - 4Y = -2 3X-Y = 8 Solve for y

If neither variable has a coefficient of 1 or -1, it might be easier to use the linear combination method. The goal of this method is to add the two equations togather to obtain an equation in one variable.

Example two
Solve the word problem by substitution.

You want to burn 380 calories during 40 minutes of exercise. you burn about 8 calories per minute running around the track and 12 calories per minute swimming. how long should you spend doing each activity?

x= minutes of running
y= Minutes of swimming

Equation One: 8x + 12y = 380
equation two: x + y = 40

Solve equation two for x

x = 40 - y

SUBSTITUTE THE VALUE OF x INTO THE ORIGINAL EQUATION TWO AND SOLVE FOR y.

Equation One: 8x + 12y = 380
Substitute 40-Y for x
Equation One: 8(40-Y) + 12y = 380
Solve
y=15

SUBSTITUTE THE VALUE OF Y INTO THE ORIGINAL EQUATION TWO AND SOLVE FOR X.

equation two: x + y = 40
substitute 15 for y
equation two: x + 15 = 40
solve
X=25

You find that the solution is (25,15).
You can check if this solution is correct by putting it back into the equations.

You find that the equation is correct. therefore you should spend 25 minutes running and 15 minutes swimming.