To solve the system of equations using the elimination method, we first write out the two equations:
- \( 5x - 9y = -10 \) (Equation 1)
- \( -9x + 6y = 18 \) (Equation 2)
Next, we will try to eliminate one variable. To do this, we can multiply the equations by appropriate values so that the coefficients of one of the variables match. Let's eliminate \( x \).
We can achieve this by making the coefficients of \( x \) in both equations the same. We will multiply Equation 1 by \( 9 \) and Equation 2 by \( 5 \):
-
\( 9(5x - 9y) = 9(-10) \) \[ 45x - 81y = -90 \quad (Equation 3) \]
-
\( 5(-9x + 6y) = 5(18) \) \[ -45x + 30y = 90 \quad (Equation 4) \]
Now we have the following system:
- \( 45x - 81y = -90 \) (Equation 3)
- \( -45x + 30y = 90 \) (Equation 4)
Next, we will add Equation 3 and Equation 4 to eliminate \( x \):
\[ (45x - 81y) + (-45x + 30y) = -90 + 90 \]
Simplifying this gives:
\[ -81y + 30y = 0 \]
Combine like terms:
\[ -51y = 0 \]
Next, solve for \( y \):
\[ y = 0 \]
Now that we have found \( y \), we can substitute \( y = 0 \) back into one of the original equations to solve for \( x \). Let's use Equation 1:
\[ 5x - 9(0) = -10 \]
This simplifies to:
\[ 5x = -10 \]
Dividing both sides by \( 5 \):
\[ x = -2 \]
Thus, the solution to the system of equations is:
\[ \boxed{(x, y) = (-2, 0)} \]
4 answers