Solve the system of equations. Enter your answer as an ordered pair.
[tex]\begin{array}{c}
4 x+5 y=14 \\
-4 x-2 y=-8
\end{array}[/tex]
Answer (1)
Add the two equations to eliminate x and solve for y .
Substitute the value of y back into one of the original equations to solve for x .
The solution to the system of equations is ( 1 , 2 ) .
Explanation
Analyze the problem We are given a system of two linear equations:
Equation 1: 4 x + 5 y = 14 Equation 2: − 4 x − 2 y = − 8
Our goal is to find the values of x and y that satisfy both equations. We will use the elimination method to solve for x and y .
Eliminate x Add Equation 1 and Equation 2 to eliminate x :
( 4 x + 5 y ) + ( − 4 x − 2 y ) = 14 + ( − 8 )
This simplifies to:
3 y = 6
Solve for y Solve for y :
y = 3 6 = 2
Substitute y into Equation 1 Substitute y = 2 into Equation 1:
4 x + 5 ( 2 ) = 14
This simplifies to:
4 x + 10 = 14
Solve for x Solve for x :
4 x = 14 − 10 = 4
x = 4 4 = 1
State the solution The solution is ( x , y ) = ( 1 , 2 ) .
Examples
Systems of equations are used in various real-life scenarios, 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 two different products 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 number of units for each product needed to break even. This helps in making informed business decisions.
