Find the solution of the system of equations:
[tex]
\begin{aligned}
-3 x-2 y & =-6 \\
7 x+6 y & =26
\end{aligned}
[/tex]
Answer (2)
Multiply the first equation by 3: − 9 x − 6 y = − 18 .
Add the modified equation to the second equation: − 2 x = 8 .
Solve for x : x = − 4 .
Substitute x into the first equation and solve for y : y = 9 . The solution is ( − 4 , 9 ) .
Explanation
Analyze the problem We are given a system of two linear equations in two variables, x and y .
Equation 1: − 3 x − 2 y = − 6 Equation 2: 7 x + 6 y = 26
Our goal is to find the values of x and y that satisfy both equations. We will use the elimination method to solve this system.
Multiply the first equation Multiply the first equation by 3 to make the coefficients of y in both equations additive inverses:
3 × ( − 3 x − 2 y ) = 3 × ( − 6 )
− 9 x − 6 y = − 18
Eliminate y Add the modified first equation to the second equation to eliminate y :
( 7 x + 6 y ) + ( − 9 x − 6 y ) = 26 + ( − 18 )
− 2 x = 8
Solve for x Solve the resulting equation for x :
x = − 2 8
x = − 4
Solve for y Substitute the value of x back into the first original equation to solve for y :
− 3 ( − 4 ) − 2 y = − 6
12 − 2 y = − 6
− 2 y = − 6 − 12
− 2 y = − 18
y = − 2 − 18
y = 9
State the solution The solution is x = − 4 and y = 9 . We can write the solution as an ordered pair ( − 4 , 9 ) .
Verify the solution Verify the solution by substituting the values of x and y into both original equations:
Equation 1: − 3 ( − 4 ) − 2 ( 9 ) = 12 − 18 = − 6 (Correct) Equation 2: 7 ( − 4 ) + 6 ( 9 ) = − 28 + 54 = 26 (Correct)
Since the solution satisfies both equations, it is the correct solution.
Final Answer The solution to the system of equations is x = − 4 and y = 9 .
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. By solving the system, they can find the break-even point, which is the number of units they need to sell to make their revenue equal to their expenses. This helps them make informed decisions about pricing, production, and marketing strategies.
The solution to the system of equations is ( − 4 , 9 ) . By using the elimination method, we eliminated one variable to solve for x and then substituted back to find y . The solution has been verified as correct in both original equations.
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