Question

Example Use substitution to solve this system of equations .y 1 3x 5 24 y 5 x 1 4 The second equation tells you Now you can find the value of y . that y 5 x 1 4, so you can substitute You can substitute 22 for x into x 1 4 for y in the first equation and either equation and solve for y . solve for x . Try using the second equation .y 1 3x 5 24 y 5 x 1 4 (x 1 4) 1 3x 5 24 4x 1 4 5 24 y 5 22 1 4 4x 5 28 y 5 2 explain why you substitute x 1 4 for y in the first equation of the system in the example .

Answer 1

To solve the system of equations using substitution, we start with the two equations:

1. \( y + 3x = 24 \) (Equation 1)
2. \( y = x + 4 \) (Equation 2)

### Step 1: Substitute
The second equation gives us a direct expression for \( y \) in terms of \( x \). We can use this expression to substitute \( y \) in the first equation.

### Explanation of the Substitution
We substitute because we want to express everything in terms of a single variable (in this case, \( x \)). By substituting \( y \) with \( x + 4 \) in Equation 1, we eliminate \( y \) and can solve solely for \( x \). This is often easier than trying to simultaneously solve two equations with two variables.

### Step 2: Substitute \( y \) in Equation 1
By substituting \( y \) from Equation 2 into Equation 1, we have:

\[
(x + 4) + 3x = 24
\]

### Step 3: Simplify and Solve for \( x \)
Now, we simplify the equation:

\[
x + 4 + 3x = 24
\]
\[
4x + 4 = 24
\]

Next, we isolate \( x \):

\[
4x = 24 - 4
\]
\[
4x = 20
\]
\[
x = 5
\]

### Step 4: Find \( y \)
Now that we have \( x \), we substitute back into Equation 2 to find \( y \):

\[
y = x + 4 = 5 + 4 = 9
\]

### Conclusion
The solution to the system of equations is \( x = 5 \) and \( y = 9 \). We substituted \( y = x + 4 \) into the first equation because it allowed us to reduce the system to a single variable, making the problem easier to solve for one variable at a time.

Thus, the final answer is:
- \( x = 5 \)
- \( y = 9 \)