Find the solution of the system of equations.
[tex]
\begin{aligned}
2 x-9 y & =14 \
-5 x+9 y & =-8
\end{aligned}
[/tex]
Answer (2)
Add the two equations to eliminate y and solve for x : x = − 2 .
Substitute the value of x into one of the equations to solve for y : y = − 2 .
The solution to the system of equations is x = − 2 and y = − 2 .
Express the solution as an ordered pair: ( − 2 , − 2 ) .
Explanation
Problem Analysis We are given a system of two linear equations with two variables, x and y . Our goal is to find the values of x and y that satisfy both equations simultaneously.
Identifying the Equations The given equations are:
Equation 1: 2 x − 9 y = 14
Equation 2: − 5 x + 9 y = − 8
Notice that the coefficients of y in the two equations are opposites ( − 9 and 9 ). This makes it convenient to eliminate y by adding the two equations.
Eliminating y and Solving for x Adding Equation 1 and Equation 2, we get:
( 2 x − 9 y ) + ( − 5 x + 9 y ) = 14 + ( − 8 )
Simplifying, we have:
2 x − 5 x − 9 y + 9 y = 14 − 8
− 3 x = 6
Dividing both sides by − 3 , we find:
x = − 3 6 = − 2
Substituting x and Solving for y Now that we have the value of x , we can substitute it into either Equation 1 or Equation 2 to solve for y . Let's use Equation 1:
2 x − 9 y = 14
Substitute x = − 2 :
2 ( − 2 ) − 9 y = 14
− 4 − 9 y = 14
Add 4 to both sides:
− 9 y = 14 + 4
− 9 y = 18
Divide both sides by − 9 :
y = − 9 18 = − 2
The Solution Therefore, the solution to the system of equations is x = − 2 and y = − 2 . We can write this as an ordered pair ( − 2 , − 2 ) .
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, suppose a company sells two products. By setting up a system of equations representing the costs and revenues associated with each product, one can determine the quantity of each product that needs to be sold to achieve profitability. This helps in making informed business decisions and optimizing resource allocation.
The solution to the given system of equations is ( − 2 , − 2 ) , found by adding the two equations to eliminate y , solving for x , and then substituting back to find y .
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