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Question 1196725: -4x+9y=9
x-3y=-6
Found 2 solutions by math_tutor2020, josgarithmetic:
Answer by math_tutor2020(3817)   (Show Source):
You can put this solution on YOUR website!
I'll go over 3 ways to solve.
Method 1) Elimination
We have 9y in the first equation and -3y in the second.
If we can get the coefficients to be equal and opposite, then we can add them to cancel out.
So let's triple everything in the second equation to go from x-3y = -6 to 3x-9y = -18
We have this equivalent system
Add the equations straight down
The x terms add to -4x+3x = -x
The y terms cancel out 9y+(-9y) = 0y = 0
The right hand sides add to -9 since 9+(-18) = -9
We're left with -x = -9 which solves to x = 9
Then plug this into any previous equation involving x and y
-4x+9y = 9
-4(9)+9y = 9
-36+9y = 9
9y = 9+36
9y = 45
y = 45/9
y = 5
The solution is (x,y) = (9,5)
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Method 2) Substitution
With this method, we look for variables where the coefficients are easy to work with. In this case, 1 is easiest since dividing by it doesn't change the number.
The equation x-3y = -6 is the same as 1x-3y = -6
Let's solve for x in this equation
x-3y = -6
x = -6+3y
Then plug this into the other equation and solve for y
-4x+9y = 9
-4(-6+3y)+9y = 9
24-12y+9y = 9
24-3y = 9
-3y = 9-24
-3y = -15
y = -15/(-3)
y = 5
Now use this to find x
x = -6+3y
x = -6+3*5
x = -6+15
x = 9
We arrive at the same solution (x,y) = (9,5)
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Method 3) Graphing
I'll use technology to graph each equation. Though you could graph each line by hand. The idea behind that is to plug in various x values to find corresponding paired y values. Then draw a straight line through each (x,y) point on the graph.
Here's the graph if you were to use Desmos
https://www.desmos.com/calculator/pipioay22g
Though you can use whichever app you prefer better. GeoGebra is another good option.
The two lines intersect at (x,y) = (9,5)
Graphing is a great way to visually confirm the answers quickly.
However, a more formal way to confirm the answers is detailed in the next section below.
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Check:
Plug x = 9 and y = 5 into the first original equation
-4x+9y = 9
-4*9+9*5 = 9
-36+45 = 9
9 = 9
Since we arrive at a true statement, this makes the original equation true when we plug in those x,y values mentioned.
Repeat similar steps for the second original equation
x-3y = -6
9-3*5 = -6
9-15 = -6
-6 = -6
Both equations are true for these x,y values.
The answer is confirmed.
This system is consistent and independent since we get one single solution.
Answer by josgarithmetic(39617) (Show Source):
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