To solve the system of equations using substitution, we start with the given equations:
- \( x + y = 4 \)
- \( y = 7x + 4 \)
Step 1: Solve for one variable
We will use the first equation to express \( y \) in terms of \( x \).
From the equation: \[ y = 4 - x \]
Step 2: Substitute into the second equation
Now we will substitute \( y \) from the first equation into the second equation.
The second equation is: \[ y = 7x + 4 \]
Substituting \( y \): \[ 4 - x = 7x + 4 \]
Step 3: Solve for \( x \)
Now we will solve the equation for \( x \).
First, we will simplify the equation: \[ 4 - x = 7x + 4 \]
Subtracting \( 4 \) from both sides: \[ -x = 7x \]
Now, add \( x \) to both sides: \[ 0 = 8x \]
Dividing both sides by \( 8 \): \[ x = 0 \]
Step 4: Substitute \( x \) back to find \( y \)
Now we can substitute \( x = 0 \) back into the first equation to find \( y \): \[ y = 4 - x \] \[ y = 4 - 0 \] \[ y = 4 \]
Conclusion
The solution to the system of equations is: \[ (x, y) = (0, 4) \]
Summary of the solution:
- From \( x + y = 4 \), we found \( y = 4 - x \).
- Substituted \( y = 4 - x \) into \( y = 7x + 4 \), leading to the equation \( 4 - x = 7x + 4 \).
- Solved for \( x \) and found \( x = 0 \).
- Substituted \( x = 0 \) back to find \( y = 4 \).
- Therefore, the solution is \( (0, 4) \).