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Math  /  Algebra

Question7x+10y=−27x+8y=−10\begin{array}{l}7 x+10 y=-2 \\ 7 x+8 y=-10\end{array}

Studdy Solution

STEP 1

1. We are given a system of two 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 both equations simultaneously.
3. We will use the elimination method to solve the system.

STEP 2

1. Set up the system of equations.
2. Eliminate one variable by subtracting the equations.
3. Solve for the remaining variable.
4. Substitute back to find the other variable.
5. Check the solution by substituting both variables back into the original equations.

STEP 3

Write down the system of equations:
\[ \begin{array}{l} 7x + 10y = -2 \quad \text{(Equation 1)} \\ 7x + 8y = -10 \quad \text{(Equation 2)} \end{array}$

STEP 4

Subtract Equation 2 from Equation 1 to eliminate x x :
(7x+10y)−(7x+8y)=−2−(−10)(7x + 10y) - (7x + 8y) = -2 - (-10)
Simplify the equation:
2y=82y = 8

STEP 5

Solve for y y :
y=82=4y = \frac{8}{2} = 4

STEP 6

Substitute y=4 y = 4 back into Equation 1 to solve for x x :
7x+10(4)=−27x + 10(4) = -2
Simplify and solve for x x :
7x+40=−27x=−2−407x=−42x=−427=−67x + 40 = -2 \\ 7x = -2 - 40 \\ 7x = -42 \\ x = \frac{-42}{7} = -6

STEP 7

Check the solution by substituting x=−6 x = -6 and y=4 y = 4 back into both original equations.
For Equation 1:
7(−6)+10(4)=−2−42+40=−2−2=−2(True)7(-6) + 10(4) = -2 \\ -42 + 40 = -2 \\ -2 = -2 \quad \text{(True)}
For Equation 2:
7(−6)+8(4)=−10−42+32=−10−10=−10(True)7(-6) + 8(4) = -10 \\ -42 + 32 = -10 \\ -10 = -10 \quad \text{(True)}
Both equations are satisfied.
The solution to the system is:
x=−6,y=4 x = -6, \quad y = 4

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