Solve the system by the substitution method.
[tex]\begin{aligned}
4 x+8 y & = -24 \\
5 x-y & = 36
\end{aligned}[/tex]
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A. The solution set is { }. (Type an ordered pair.)
B. There are infinitely many solutions.
C. There is no solution.
Answer (2)
Solve the second equation for y : y = 5 x − 36 .
Substitute this expression for y into the first equation: 4 x + 8 ( 5 x − 36 ) = − 24 .
Simplify and solve for x : x = 6 .
Substitute the value of x back into the equation y = 5 x − 36 to find the value of y : y = − 6 . The solution is ( 6 , − 6 ) .
Explanation
Analyze the problem We are given the following system of equations: 4 x + 8 y = − 24 5 x − y = 36 We will solve this system using the substitution method.
Solve for y First, we solve the second equation for y :
5 x − y = 36 y = 5 x − 36
Substitute into first equation Now, we substitute this expression for y into the first equation: 4 x + 8 ( 5 x − 36 ) = − 24
Solve for x Next, we simplify and solve for x :
4 x + 40 x − 288 = − 24 44 x = 264 x = 44 264 x = 6
Solve for y Now, we substitute the value of x back into the equation y = 5 x − 36 to find the value of y :
y = 5 ( 6 ) − 36 y = 30 − 36 y = − 6
Check the solution The solution is ( 6 , − 6 ) . We check the solution by substituting the values of x and y into both original equations: 4 ( 6 ) + 8 ( − 6 ) = 24 − 48 = − 24 5 ( 6 ) − ( − 6 ) = 30 + 6 = 36 Both equations are satisfied.
Final Answer Therefore, the solution set is ( 6 , − 6 ) .
Examples
Systems of equations are used in many real-world applications. For example, businesses use systems of equations to model supply and demand, to optimize production, and to determine pricing strategies. In science, systems of equations are used to model chemical reactions, to analyze electrical circuits, and to predict the behavior of complex systems. In everyday life, we use systems of equations to solve problems such as determining the best combination of items to purchase within a budget or to calculate distances and speeds when traveling.
The solution set of the given system of equations is ( 6 , − 6 ) . This solution was found by substituting the expression for y from the second equation into the first equation, simplifying, and then solving for x and y . Both original equations were verified to be true with these values.
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