SOLUTION: Choosing A Method: Solve the linear system. 8x + 9y = 42 6x - y = 16 I can get my first equation to equal 42 with the coordinates (2/5/23,2/16/23) and (1/20/31,3/19/93)

Question 132034This question is from textbook
: Choosing A Method:
Solve the linear system.

8x + 9y = 42
6x - y = 16

I can get my first equation to equal 42 with the coordinates (2/5/23,2/16/23) and (1/20/31,3/19/93)
But I can't get either of those coordinates to fit in the bottom equation. This question is from textbook

Answer by bucky(2189) (Show Source): You can put this solution on YOUR website!
Given the set of equations:
.
8x + 9y = 42
6x - y = 16
.
There are three ways you can solve these two equations to get a common solution.
.
One way is to graph both equations and find the point where the two linear graphs cross.
That (x, y) intersection point will give you the values that are common to both equations.
.
Another way to do the problem is to solve one of the equations for one of the variables
and substitute the equivalent value into the other equation, and solve it.
.
Let's do that method. Note that in the bottom equation, you have a "-y" term. If you subtract
6x from both sides of this bottom equation you have:
.
-y = -6x + 16
.
Then multiply both sides (all terms) by -1 and the equation becomes:
.
y = 6x - 16
.
Next take the right side of this equation (6x - 16) and substitute it for y in the original top
equation. When you make that substitution in the original top equation you get:
.
8x + 9(6x - 16) = 42
.
Multiply the 9 times each of the terms inside the parentheses and you get:
.
8x + 54x - 144 = 42
.
Combine the two terms containing x ... (8x + 54x = 62x) ... and the equation becomes:
.
62x - 144 = 42
.
Get rid of the -144 on the left side by adding 144 to both sides to get:
.
62x = 186
.
Solve for x by dividing both sides of this equation by 62 and the result is:
.
x = 186/62 = 3
.
Now that you know x = 3, you can solve for y by returning to either of the original equations
and substituting 3 for x. Then solve for y. Let's return to the bottom of the two original equations:
.
6x - y = 16
.
Substitute 3 for x and the equation becomes:
.
6*3 - y = 16
.
Multiplying out 6*3 and you get:
.
18 - y = 16
.
Subtract 18 from both sides:
.
-y = -2
.
Multiply both sides by -1:
.
y = 2
.
So the common solution to both equations is x = 3 and y = 2. This tells you that the (x, y)
point where the two graphs cross is (3, 2). It also tells you that if you substitute 3
for x and 2 for y in the two original equations, the left side of each equation should equal the
right side.
.
A third way to work this problem is by the method of variable elimination. The way to do this
is to get a term in the top equation and the corresponding term in the bottom equation to be
equal in value. Then you vertically add or subtract that term to get rid of it. Let's do it.
.
Start with the two original equations:
.
8x + 9y = 42
6x - y = 16
.
Multiply the bottom equation (both sides all terms) by 9 to make the equation set become:
.
8x + 9y = 42
54x -9y = 144
.
Now if you add the two equations in vertical columns, note that the +9y and the -9y cancel
each other out and the sum of the two equations vertically becomes:
.
8x + 9y = 42
54x -9y = 144
--------------
62x + 0 = 186
.
Which simplifies to:
.
62x = 186
.
Divide both sides by 62, just as we did in the substitution method and you again get the
answer x = 3. Solve for y by going back to the original equations and in one of them replace
x by 3 and solve for y.
.
Hope this helps you to see the ways you can do this problem.
.
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