Questions: Solve the system of equations by the elimination method. 9x + 9y = 18 -3x - 4y = -8 no solution (0,2) (0,3) (-1,3)

Transcript text: Solve the system of equations by the elimination method. \[ \begin{array}{l} 9 x+9 y=18 \\ -3 x-4 y=-8 \end{array} \] no solution $(0,2)$ $(0,3)$ $(-1,3)$

Solution

Solution Steps

To solve the system of equations using the elimination method, we need to eliminate one of the variables by adding or subtracting the equations. We can multiply the equations by appropriate constants to make the coefficients of one of the variables equal in magnitude but opposite in sign. Then, we add or subtract the equations to eliminate that variable and solve for the remaining variable. Finally, we substitute back to find the other variable.

Step 1: Write the System of Equations

We start with the given system of equations: \[ \begin{array}{l} 9x + 9y = 18 \\ -3x - 4y = -8 \end{array} \]

Step 2: Eliminate One Variable

To eliminate one of the variables, we can multiply the second equation by 3 to make the coefficient of $x$ in both equations equal in magnitude but opposite in sign: \[ \begin{array}{l} 9x + 9y = 18 \\ -9x - 12y = -24 \end{array} \]

Step 3: Add the Equations

Next, we add the two equations to eliminate $x$: \[ (9x + 9y) + (-9x - 12y) = 18 + (-24) \] \[ 9y - 12y = -6 \] \[ -3y = -6 \]

Step 4: Solve for $y$

Solving for $y$: \[ y = \frac{-6}{-3} = 2 \]

Step 5: Substitute $y$ Back into One of the Original Equations

We substitute $y = 2$ back into the first equation to solve for $x$: \[ 9x + 9(2) = 18 \] \[ 9x + 18 = 18 \] \[ 9x = 0 \] \[ x = 0 \]