What is the solution to the system of equations?
[tex]\begin{array}{l}
y=-3 x-2 \\
5 x+2 y=15
\end{array}[/tex]
A. (-40,19)
B. (-19,55)
C. (19,-40)
D. (55,-19)
Answer (1)
Substitute the first equation into the second equation: 5 x + 2 ( − 3 x − 2 ) = 15 .
Solve for x: 5 x − 6 x − 4 = 15 ⇒ − x = 19 ⇒ x = − 19 .
Substitute x = -19 into the first equation: y = − 3 ( − 19 ) − 2 .
Solve for y: y = 57 − 2 = 55 . The solution is ( − 19 , 55 ) .
Explanation
Analyze the problem We are given a system of two linear equations:
y = − 3 x − 2
5 x + 2 y = 15
We need to find the solution (x, y) that satisfies both equations. We can use the substitution method to solve this system.
Substitution Substitute the first equation into the second equation:
5 x + 2 ( − 3 x − 2 ) = 15
Solve for x Simplify and solve for x:
5 x − 6 x − 4 = 15
− x − 4 = 15
− x = 19
x = − 19
Solve for y Substitute the value of x back into the first equation to find y:
y = − 3 ( − 19 ) − 2
Calculate y Calculate the value of y:
y = 57 − 2
y = 55
State the solution The solution to the system of equations is x = -19 and y = 55. Therefore, the solution is (-19, 55).
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. For instance, if a company wants to know how many units of a product they need to sell to cover their costs, they can set up a system of equations to represent their revenue and expenses. Solving this system will give them the break-even point, which is the number of units they need to sell to make a profit.
