Questions: Solve the system of equations by using the addition method. 2x - 18y = 10 6x + 29y = 30 The solution set is .

Solution Steps

To solve the given system of linear equations using the addition method, we need to eliminate one of the variables by adding or subtracting the equations. First, we can multiply the equations by suitable numbers 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, solving for the other variable. Finally, we substitute back to find the value of the eliminated variable.

We start with the given system of equations: \[ \begin{align*}

1. & \quad 2x - 18y = 10 \\
2. & \quad 6x + 29y = 30 \end{align*} \]

Using the addition method, we can manipulate the equations to eliminate one variable. We can multiply the first equation by 3 to align the coefficients of \(x\): \[ 3(2x - 18y) = 3(10) \implies 6x - 54y = 30 \] Now we have: \[ \begin{align*}

1. & \quad 6x - 54y = 30 \\
2. & \quad 6x + 29y = 30 \end{align*} \]

Next, we subtract the second equation from the first: \[ (6x - 54y) - (6x + 29y) = 30 - 30 \] This simplifies to: \[ -54y - 29y = 0 \implies -83y = 0 \implies y = 0 \]

Now that we have \(y = 0\), we substitute this value back into one of the original equations to find \(x\). Using the first equation: \[ 2x - 18(0) = 10 \implies 2x = 10 \implies x = 5 \]

Final Answer