solve the following system of linear equations by elimination. verify your solution.
1. -4x-9y=9 2. x-3y=-6
- 4x-9y=9
+1x-3y=-6 multiply this by -3
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- 4x-9y= 9
-3x+9y=18
======================= now add
- 7 x = 27
x = -27/7
now y
+1x-3y=-6
so 3 y = 6 + x
3 y =42/7 -27/7 = 15/7
y = 5/7
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check
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- 4x-9y=9 ?
-4 (-27/7) - 9 (5/7) = 9 ????
(108 - 45)/7
63/7
9 finest kind !!!
To solve the system of linear equations by elimination, we need to eliminate one variable from either equation.
Let's start by eliminating the variable "x".
To do this, we will multiply equation (2) by 4 to make the coefficients of "x" in the two equations opposite each other:
(2) 4(x - 3y) = 4(-6)
Simplifying equation (2):
4x - 12y = -24
Now, we will add this newly obtained equation to equation (1):
-4x - 9y + 4x - 12y = 9 - 24
Simplifying this equation:
-21y = -15
Dividing both sides of the equation by -21:
y = (-15)/(-21)
y = 5/7
Now that we have the value of "y", we can substitute it into either equation to find the value of "x". Let's substitute it into equation (1):
-4x - 9(5/7) = 9
-4x - 45/7 = 9
To simplify the equation, let's get rid of the fraction:
(-4x)(7) - 45 = 9(7)
-28x - 45 = 63
Adding 45 to both sides of the equation:
-28x = 63 + 45
-28x = 108
Dividing both sides of the equation by -28:
x = 108/(-28)
x = -54/14
x = -27/7
Therefore, the solution to the system of linear equations is:
x = -27/7, y = 5/7.
To verify this solution, substitute the values of x and y into the original equations and check if they are satisfied:
Equation (1): -4x - 9y = 9
(-4)(-27/7) - 9(5/7) = 9
108/7 - 45/7 = 9
(108 - 45)/7 = 9
63/7 = 9
9 = 9 (True)
Equation (2): x - 3y = -6
(-27/7) - 3(5/7) = -6
(-27 - 15)/7 = -6
(-42)/7 = -6
-6 = -6 (True)
Both equations are true when we substitute the values of x = -27/7 and y = 5/7 into them. Therefore, our solution is verified.
To solve a system of linear equations by elimination, we need to manipulate the equations in a way that allows us to eliminate one variable when we add or subtract the two equations.
Let's start with the given system:
1. -4x - 9y = 9 --- Equation (1)
2. x - 3y = -6 --- Equation (2)
First, we want to eliminate one variable, so let's eliminate the x variable. To do that, we will multiply equation (2) by 4 and equation (1) by 1, so that the coefficients of x in both equations will be opposite in sign.
Multiply equation (2) by 4:
4(x - 3y) = 4(-6)
4x - 12y = -24 --- Equation (3)
Next, we can add equation (1) and (3) to eliminate the x variable:
(-4x - 9y) + (4x - 12y) = (9) + (-24)
Combining like terms, we have:
-4x + 4x - 9y - 12y = 9 - 24
-21y = -15
Now, we can solve for y by dividing both sides of the equation by -21:
y = -15 / -21
y = 5 / 7
We have found the value of y. To find the value of x, we substitute the value of y into either equation (1) or (2). Let's use equation (2):
x - 3(5/7) = -6
x - 15/7 = -6
To get rid of the fraction, we multiply through by the common denominator, which is 7:
7x - 15 = -42
Next, we solve for x by isolating the x variable:
7x = -42 + 15
7x = -27
Dividing both sides by 7, we find:
x = -27/7
So the solution to the system of equations is x = -27/7 and y = 5/7.
To verify our solution, we substitute these values back into the original equations and check if they hold true.