Solve the system of equations. Enter your answer as an ordered pair.
[tex]\begin{array}{l}
5 x+5 y=20 \\
-5 x+4 y=7
\end{array}[/tex]
Solve the system of equations. Enter your answer as an ordered pair.
[tex]\begin{array}{c}
10 x-8 y=-2 \\
9 x+8 y=59
\end{array}[/tex]
Answer (2)
Solve the first system by adding the equations to eliminate x , then solve for y and substitute back to find x , resulting in ( 1 , 3 ) .
Solve the second system by adding the equations to eliminate y , then solve for x and substitute back to find y , resulting in ( 3 , 4 ) .
The solutions to the systems of equations are ( 1 , 3 ) and ( 3 , 4 ) .
Explanation
Problem Analysis We are given two systems of linear equations, and we need to solve each system for x and y . The solutions should be expressed as ordered pairs ( x , y ) .
Solving the First System For the first system of equations:
5 x + 5 y = 20 − 5 x + 4 y = 7
We can use the elimination method. Adding the two equations eliminates x :
( 5 x + 5 y ) + ( − 5 x + 4 y ) = 20 + 7 9 y = 27 y = 9 27 = 3
Now, substitute y = 3 into the first equation:
5 x + 5 ( 3 ) = 20 5 x + 15 = 20 5 x = 5 x = 1
So the solution for the first system is ( 1 , 3 ) .
Solving the Second System For the second system of equations:
10 x − 8 y = − 2 9 x + 8 y = 59
We can use the elimination method again. Adding the two equations eliminates y :
( 10 x − 8 y ) + ( 9 x + 8 y ) = − 2 + 59 19 x = 57 x = 19 57 = 3
Now, substitute x = 3 into the first equation:
10 ( 3 ) − 8 y = − 2 30 − 8 y = − 2 − 8 y = − 32 y = − 8 − 32 = 4
So the solution for the second system is ( 3 , 4 ) .
Final Answer Therefore, the solutions to the systems of equations are ( 1 , 3 ) and ( 3 , 4 ) .
Examples
Systems of equations are useful in many real-world scenarios, such as determining the break-even point for a business. For example, suppose a company has fixed costs of $10,000 and variable costs of $5 per unit. If the company sells each unit for 15 , w ec an se t u p a sys t e m o f e q u a t i o n s t o f in d t h e n u mb ero f u ni t s t h eco m p an y n ee d s t ose llt o b re ak e v e n . L e t x b e t h e n u mb ero f u ni t s an d y b e t h e t o t a l cos t / re v e n u e . T h ecos t e q u a t i o ni s y = 5x + 10000 an d t h ere v e n u ee q u a t i o ni s y = 15x$. Solving this system gives the break-even point.
The solutions for the systems of equations are (1, 3) for the first system and (3, 4) for the second system. Each solution was found by applying the elimination method and substituting values back into the equations. This process provides the correct values for x and y as ordered pairs.
;
