Solve the system with the addition method:
$\left\{\begin{array}{l}
-9 x+8 y=43 \\
3 x-8 y=-25
\end{array}\right.$
Answer: $(x, y)=($ $\square$ , $\square$)
Enter your answers as integers or as reduced fraction(s) in the form $A / B$.
Answer (2)
We solved the system of equations by first eliminating y, leading to the value of x as -3. We then substituted x back into one of the equations, which gave us y as 2. The final solution for the system is (x, y) = (-3, 2).
;
Add the two equations to eliminate y : − 6 x = 18 .
Solve for x : x = − 3 .
Substitute x = − 3 into one of the equations and solve for y : y = 2 .
The solution is ( − 3 , 2 ) .
Explanation
Analyze the problem and data We are given the following system of equations:
{ − 9 x + 8 y = 43 3 x − 8 y = − 25
We will use the addition method to solve for x and y .
Eliminate y and solve for x Add the two equations to eliminate y :
( − 9 x + 8 y ) + ( 3 x − 8 y ) = 43 + ( − 25 ) − 6 x = 18
Divide both sides by − 6 to solve for x :
x = − 6 18 = − 3
So, x = − 3 .
Solve for y Substitute the value of x into the second equation to solve for y :
3 x − 8 y = − 25 3 ( − 3 ) − 8 y = − 25 − 9 − 8 y = − 25
Add 9 to both sides:
− 8 y = − 25 + 9 − 8 y = − 16
Divide both sides by − 8 to solve for y :
y = − 8 − 16 = 2
So, y = 2 .
State the solution The solution to the system of equations is ( x , y ) = ( − 3 , 2 ) .
Examples
Systems of equations are used in various real-world applications, such as determining the break-even point for a business, calculating the optimal mix of ingredients in a recipe, or modeling supply and demand in economics. In this case, solving the system of equations helps us find the values of x and y that satisfy both equations simultaneously, which can be useful in scenarios where two variables are related by two different linear relationships.
