What is the solution to this system of equations?
$\begin{array}{l}
y=2-x \\
3 x-2 y=16
\end{array}$
(4,-2)
(0,2)
(5,-3)
(-4,2)
Answer (1)
Substitute each given (x, y) pair into both equations.
Check if both equations are true for a given pair.
The point (4, -2) satisfies both equations: y = 2 − x and 3 x − 2 y = 16 .
Therefore, the solution to the system of equations is ( 4 , − 2 ) .
Explanation
Problem Analysis We are given a system of two linear equations and four possible solutions. Our goal is to find the solution that satisfies both equations. We will test each option by substituting the x and y values into both equations.
Testing the First Option Let's test the first option, (4, -2). We substitute x = 4 and y = -2 into the equations:
Equation 1: y = 2 − x becomes − 2 = 2 − 4 , which simplifies to − 2 = − 2 . This is true. Equation 2: 3 x − 2 y = 16 becomes 3 ( 4 ) − 2 ( − 2 ) = 16 , which simplifies to 12 + 4 = 16 , or 16 = 16 . This is also true.
Since (4, -2) satisfies both equations, it is the solution.
Conclusion Since we found a solution in the first option, we don't need to test the other options.
Examples
Systems of equations are used in various real-life scenarios, such as determining the break-even point for a business. For example, if a company has fixed costs and variable costs per unit, and they sell each unit at a certain price, a system of equations can be set up to find the number of units they need to sell to cover their costs and start making a profit. Another example is in mixture problems, where you need to find the amount of each ingredient to mix to get a desired concentration or quantity.
