Which of the following points is a solution to the system
[tex]\begin{array}{l}
3 x+2 y=17 \\
7 x-5 y=1
\end{array}[/tex]
Answer (2)
Multiply the first equation by 5 and the second equation by 2 to eliminate y .
Add the modified equations to solve for x : 29 x = 87 , which gives x = 3 .
Substitute x = 3 into the first original equation: 3 ( 3 ) + 2 y = 17 .
Solve for y : 2 y = 8 , which gives y = 4 . The solution is ( 3 , 4 ) .
Explanation
Analyze the problem We are given a system of two linear equations:
3 x + 2 y = 17
7 x − 5 y = 1
Our goal is to find the values of x and y that satisfy both equations simultaneously. We can solve this system using either the substitution or elimination method.
Eliminate y Let's use the elimination method. Multiply the first equation by 5 and the second equation by 2 to eliminate y :
( 3 x + 2 y ) × 5 = 17 × 5 ⇒ 15 x + 10 y = 85 ( 7 x − 5 y ) × 2 = 1 × 2 ⇒ 14 x − 10 y = 2
Solve for x Now, add the two new equations to eliminate y :
( 15 x + 10 y ) + ( 14 x − 10 y ) = 85 + 2 29 x = 87
Calculate x Divide both sides by 29 to solve for x :
x = 29 87 = 3
Substitute x into the first equation Substitute the value of x = 3 back into the first original equation to solve for y :
3 ( 3 ) + 2 y = 17 9 + 2 y = 17 2 y = 17 − 9 2 y = 8
Calculate y Divide both sides by 2 to solve for y :
y = 2 8 = 4
State the solution Therefore, the solution to the system of equations is x = 3 and y = 4 . The point that satisfies both equations is ( 3 , 4 ) .
Examples
Systems of equations are incredibly useful in real life. Imagine you're trying to figure out how much to charge for tickets to a school play. You need to cover your costs, but also want to make some profit. By setting up a system of equations, where one equation represents your costs and another represents your revenue, you can solve for the ticket price that meets both goals. This is just one example; systems of equations are used in economics, engineering, and many other fields to model and solve problems with multiple variables and constraints.
The solution to the system of equations is the point ( 3 , 4 ) , which satisfies both equations. After applying the elimination method, we determined that x = 3 and y = 4 . This shows how to find the intersection of two lines represented by the equations.
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