1. Solve this system of equations
and check:
x - 2y = 14
x + 3y = 9
a. First, be sure that the variables are "lined up" under
one another. In this problem, they are already "lined up".
x - 2y = 14
x + 3y = 9
b. Decide which variable ("x" or "y") will be easier to
eliminate. In order to eliminate a variable, the numbers in
front of them (the coefficients) must be the same or
negatives of one another. Looks like "x" is the easier
variable to eliminate in this problem since the x's already
have the same coefficients.
x - 2y = 14
x + 3y = 9
x - 2y = 14
- (x + 3y = 9)
c. Now, in this problem we need to subtract to eliminate
the "x" variable. Subtract ALL of the sets of lined up
terms.
(Remember: when you subtract signed numbers, you
change the signs and follow the rules for adding signed
numbers.)
x - 2y = 14
-x - 3y = - 9
- 5y = 5
d. Solve this simple equation.
-5y = 5
y = -1
e. Plug "y = -1" into either of the ORIGINAL equations
x - 2y = 14
x - 2(-1) = 14
x + 2 = 14
x = 12
to get the value for "x".
x - 2y = 14
f.
Check:
substitute x = 12 and y = -1 into BOTH
ORIGINAL equations. If these answers are correct,
BOTH equations will be TRUE!
12 - 2(-1) = 14
12 + 2 = 14
14 = 14 (check!)
x + 3y = 9
12 + 3(-1) = 9
12 - 3 = 9
9 = 9 (check!)
There is no stopping us now!
Let's try a harder problem....
2. Solve this system of equations
and check:
4x + 3y = -1
5x + 4y = 1
a. You can probably see the dilemma with this problem
4x + 3y = -1
5x + 4y = 1
right away. Neither of the variables have the same (or
negative) coefficients to eliminate. Yeek!
b. In this type of situation, we must MAKE the
coefficients the same (or negatives) by multiplication.
You can MAKE either the "x" or the "y" coefficients the
same. Pick the easier numbers. In this problem, the "y"
variables will be changed to the same coefficient by
multiplying the top equation by 4 and the bottom
equation by 3.
Remember:
* you can multiply the two differing coefficients to
obtain the new coefficient if you cannot think of another
smaller value that will work.
* multiply EVERY element in each equation by your
adjustment numbers.
4(4x + 3y = -1)
3(5x + 4y = 1)
16x + 12y = -4
15x + 12y = 3
c. Now, in this problem we need to subtract to
eliminate the "y" variable.
(Remember: when you subtract signed numbers, you
change the signs and follow the rules for adding signed
numbers.)
d. Plug "x = -7" into either of the ORIGINAL
equations to get the value for "y".
e.
Check:
substitute x = -7 and y = 9 into BOTH
ORIGINAL equations. If these answers are correct,
BOTH equations will be TRUE!
16x + 12y = -4
-15x - 12y = - 3
x
=-7
5x + 4y = 1
5(-7) + 4y = 1
-35 + 4y = 1
4y = 36
y=9
4x + 3y = -1
4(-7) +3(9) = -1
-28 + 27 = -1
-1 = -1 (check!)
5x + 4y = 1
5(-7) + 4(9) = 1
-35 + 36 = 1
1 = 1 (check!)
Let's finish with an addition method problem:
3. Solve this system of equations
and check:
4x - y = 10
2x = 12 - 3y
a. First, be sure that the variables are "lined up" under
4x - y = 10
2x + 3y = 12
one another. The second equation was rearranged so that
the variables would "line up" with those in the first
equation.
b. Decide which variable ("x" or "y") will be easier to
eliminate. In this problem, we must MAKE EITHER the
"x" or the "y" coefficients the same. The "y" variable is
being used here. Multiplying by 3 will give the "y"
variables negative coefficients. (Yes, -3 could also have
been used.)
c. Now, add to eliminate the "y" variable.
3(4x - y = 10)
2x + 3y = 12
12x - 3y = 30
2x + 3y = 12
12x - 3y = 30
2x + 3y = 12
14x
d. Solve this simple equation.
= 42
14x = 42
x=3
e. Plug "x = 3" into either of the ORIGINAL equations
to get the value for "y".
4x - y = 10
4(3) - y = 10
12 - y = 10
-y = -2
y=2
f.
Check:
substitute x = 3 and y = 2 into BOTH
ORIGINAL equations. If these answers are correct,
BOTH equations will be TRUE!
4x - y = 10
4(3) - 2 = 10
12 - 2 = 10
10 = 10 (check!)
2x = 12 - 3y
2(3) = 12 - 3(2)
6 = 12 - 6
6 = 6 (check!)