Solve the system with the addition method:
$\left\{\begin{array}{lll}
-2 x-5 y & = & 2 \\
x-9 y & = & -24
\end{array}\right.$
Answer: $(x, y)=($ $\square$, $\square$ )
Answer (1)
To solve the system of equations using the addition method:
Multiply the second equation by 2 to eliminate x .
Add the 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 ( − 6 , 2 ) .
Explanation
Analyze the problem We are given a system of two linear equations: − 2 x − 5 y = 2 x − 9 y = − 24 Our goal is to solve this system for x and y using the addition method. This involves manipulating the equations so that when we add them, one of the variables is eliminated.
Multiply the second equation by 2 To eliminate x , we can multiply the second equation by 2: 2 ( x − 9 y ) = 2 ( − 24 ) 2 x − 18 y = − 48 Now we have the following system: − 2 x − 5 y = 2 2 x − 18 y = − 48
Add the equations Add the two equations: ( − 2 x − 5 y ) + ( 2 x − 18 y ) = 2 + ( − 48 ) − 23 y = − 46
Solve for y Solve for y :
y = − 23 − 46 y = 2
Solve for x Substitute the value of y into one of the original equations to solve for x . Let's use the second equation: x − 9 y = − 24 x − 9 ( 2 ) = − 24 x − 18 = − 24 x = − 24 + 18 x = − 6
Verify the solution So the solution is x = − 6 and y = 2 . We can check our solution by substituting these values into both original equations: − 2 x − 5 y = 2 − 2 ( − 6 ) − 5 ( 2 ) = 12 − 10 = 2 x − 9 y = − 24 − 6 − 9 ( 2 ) = − 6 − 18 = − 24 Both equations are satisfied.
Final Answer The solution to the system of equations is ( x , y ) = ( − 6 , 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. In this case, solving a system of linear equations helps us find the exact values that satisfy both equations simultaneously, providing a unique solution to the problem. Understanding how to solve systems of equations is a fundamental skill in mathematics and has practical applications in many fields.
