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Math  /  Algebra

Question13. {6x−3y=−214x=3y−9\left\{\begin{array}{l}6 x-3 y=-21 \\ 4 x=3 y-9\end{array}\right.

Studdy Solution

STEP 1

1. We have a system of linear equations with two variables, x x and y y .
2. We will solve the system using the substitution or elimination method.

STEP 2

1. Simplify and rearrange the equations if necessary.
2. Use substitution or elimination to solve for one variable.
3. Substitute back to find the other variable.
4. Verify the solution by plugging the values back into the original equations.

STEP 3

First, let's write down the system of equations clearly:
1. 6x−3y=−21 6x - 3y = -21
2. 4x=3y−9 4x = 3y - 9

STEP 4

Let's solve the second equation for y y to use substitution:
4x=3y−9 4x = 3y - 9
Add 9 to both sides:
4x+9=3y 4x + 9 = 3y
Divide every term by 3 to solve for y y :
y=4x+93 y = \frac{4x + 9}{3}

STEP 5

Substitute the expression for y y from Step 2 into the first equation:
6x−3(4x+93)=−21 6x - 3\left(\frac{4x + 9}{3}\right) = -21
Simplify by distributing and canceling the 3:
6x−(4x+9)=−21 6x - (4x + 9) = -21
Combine like terms:
6x−4x−9=−21 6x - 4x - 9 = -21
2x−9=−21 2x - 9 = -21
Add 9 to both sides:
2x=−12 2x = -12
Divide by 2:
x=−6 x = -6

STEP 6

Substitute x=−6 x = -6 back into the expression for y y :
y=4(−6)+93 y = \frac{4(-6) + 9}{3}
Calculate:
y=−24+93 y = \frac{-24 + 9}{3}
y=−153 y = \frac{-15}{3}
y=−5 y = -5

STEP 7

Verify the solution by substituting x=−6 x = -6 and y=−5 y = -5 back into the original equations:
First equation: 6(−6)−3(−5)=−21 6(-6) - 3(-5) = -21 −36+15=−21 -36 + 15 = -21 −21=−21 -21 = -21 (True)
Second equation: 4(−6)=3(−5)−9 4(-6) = 3(-5) - 9 −24=−15−9 -24 = -15 - 9 −24=−24 -24 = -24 (True)
Both equations are satisfied, so the solution is correct.
The solution to the system is x=−6 x = -6 and y=−5 y = -5 .

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