Solve the system of equations using elimination: x - 5y = -9 and -5x + 8y = -6.
Understand the Problem
The question is asking to solve a system of equations using the elimination method. This involves manipulating the equations to eliminate one of the variables, allowing for the remaining variable to be solved easily.
Answer
The solution is \(x = 6\) and \(y = 3\).
Answer for screen readers
The solution to the system of equations is (x = 6) and (y = 3).
Steps to Solve
-
Write down the equations Start with the given system of equations: [ \begin{align*}
-
& \quad x - 5y = -9 \quad (1) \
-
& \quad -5x + 8y = -6 \quad (2) \end{align*} ]
-
Eliminate one variable To eliminate (x), multiply the first equation by 5: [ 5(x - 5y) = 5(-9) \implies 5x - 25y = -45 \quad (3) ]
-
Add the equations Now add equation (2) and equation (3): [ (-5x + 8y) + (5x - 25y) = -6 - 45 ] This simplifies to: [ -17y = -51 ]
-
Solve for (y) Divide both sides by -17: [ y = \frac{-51}{-17} = 3 ]
-
Substitute (y) back into one of the original equations Using equation (1): [ x - 5(3) = -9 \implies x - 15 = -9 ] Add 15 to both sides: [ x = -9 + 15 = 6 ]
The solution to the system of equations is (x = 6) and (y = 3).
More Information
In this problem, we used the elimination method to solve a system of linear equations. The values found for (x) and (y) indicate the point of intersection of these two lines on a graph.
Tips
- Not properly aligning the equations when adding them.
- Forgetting to multiply both sides of the equation when eliminating a variable.
- Miscalculating during substitution, especially when handling negative values.
AI-generated content may contain errors. Please verify critical information
