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Math  /  Algebra

QuestionSolve the following system of equations and fill in the values below: 40x+5y=−2015x−3y=1255x+2y=−8\begin{array}{l} 40 x+5 y=-20 \\ 15 x-3 y=12 \\ 55 x+2 y=-8 \end{array}
The solution is x= Blank 1‾\mathrm{x}=\underline{\text { Blank } 1} and y=\mathrm{y}= Blank 2 .
Blank 1 Add your answer

Studdy Solution

STEP 1

1. We are given a system of three linear equations with two variables, x x and y y .
2. The goal is to find the values of x x and y y that satisfy all three equations simultaneously.
3. The system might be overdetermined, meaning it could have no solution, one solution, or infinitely many solutions.

STEP 2

1. Simplify the equations if possible.
2. Use the elimination or substitution method to solve the system.
3. Verify the solution by substituting back into the original equations.

STEP 3

First, let's simplify each equation if possible.
Equation 1: 40x+5y=−20 40x + 5y = -20
Divide the entire equation by 5:
8x+y=−4 8x + y = -4
Equation 2: 15x−3y=12 15x - 3y = 12
Divide the entire equation by 3:
5x−y=4 5x - y = 4
Equation 3: 55x+2y=−8 55x + 2y = -8
No simplification needed for this equation.

STEP 4

Now, let's use the elimination method to solve the system. We'll start with the first two simplified equations:
1. 8x+y=−4 8x + y = -4
2. 5x−y=4 5x - y = 4
Add the two equations to eliminate y y :
(8x+y)+(5x−y)=−4+4 (8x + y) + (5x - y) = -4 + 4
This simplifies to:
13x=0 13x = 0
Solve for x x :
x=0 x = 0

STEP 5

Now that we have x=0 x = 0 , substitute it back into one of the simplified equations to find y y . Let's use the first simplified equation:
8(0)+y=−4 8(0) + y = -4
This simplifies to:
y=−4 y = -4

STEP 6

Verify the solution x=0 x = 0 and y=−4 y = -4 by substituting back into the original equations.
Original Equation 1: 40(0)+5(−4)=−20 40(0) + 5(-4) = -20
0−20=−20 0 - 20 = -20 (True)
Original Equation 2: 15(0)−3(−4)=12 15(0) - 3(-4) = 12
0+12=12 0 + 12 = 12 (True)
Original Equation 3: 55(0)+2(−4)=−8 55(0) + 2(-4) = -8
0−8=−8 0 - 8 = -8 (True)
All equations are satisfied.
The solution is x=0‾ x = \underline{0} and y=−4‾ y = \underline{-4} .

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