Solve the system of equations using Substitution. Make sure to explain what you are doing.

Answer 1

Of course! To solve a system of equations using substitution, we can follow these general steps:

1. Solve one of the equations for one of the variables.
2. Substitute that expression into the other equation.
3. Solve the new equation for the remaining variable.
4. Substitute back to find the other variable.

Let's work with a sample system of equations:

1. \( y = 2x + 3 \) (Equation 1)
2. \( 3x - 4y = -6 \) (Equation 2)

Step 1: Solve one equation for one variable

Here, we will use Equation 1, where \( y \) is already isolated:

\[ y = 2x + 3 \]

Step 2: Substitute this expression into the other equation

Now that we have \( y \) in terms of \( x \), we can substitute it into Equation 2:

\[ 3x - 4(2x + 3) = -6 \]

Step 3: Simplify and solve for \( x \)

Distributing \(-4\) to the terms in parentheses:

\[ 3x - 8x - 12 = -6 \]

Combine like terms:

\[ -5x - 12 = -6 \]

Now, add \( 12 \) to both sides:

\[ -5x = 6 \]

Now, divide by \(-5\):

\[ x = -\frac{6}{5} \]

Step 4: Substitute back to find \( y \)

Now that we have \( x \), we can substitute this value back into Equation 1 to find \( y \):

\[ y = 2(-\frac{6}{5}) + 3 \]

Calculating this gives:

\[ y = -\frac{12}{5} + 3 \] \[ y = -\frac{12}{5} + \frac{15}{5} = \frac{3}{5} \]

Final solution

The solution to the system of equations is:

\[ x = -\frac{6}{5}, \quad y = \frac{3}{5} \]

In conclusion, the values of \( x \) and \( y \) that satisfy both equations are:

\[ \left( -\frac{6}{5}, \frac{3}{5} \right) \]