Question

Answer the questions and complete the steps to solve the system of equations below.
3x + 2y = 16
y=x+3
a) Which variable should you substitute for in the equation 3x + 2y = 16?
choose your answer...
b) What value can be substituted for that variable? choose your answer...
c) Rewrite the equation with the substitution. choose your answer...
choose your answer...
3(x+3) + 2y =16
• x into the equation and solve for y.
3x + 2(x+3) = 16

Answer 1

Answer the questions and complete the steps to solve the system of equations below.
3x + 2y = 16
y =x+3
a) Which variable should you substitute for in the equation 3x + 2y = 16?
choose your answer...
b) What value can be substituted for that variable? choose your answer... V
c) Rewrite the equation with the substitution. choose your answer...
d) Solve the equation for x.
e) Substitute that value for x into the equation and solve for y.
f) Use substitution to check your solution for the system.

Answer 2

To solve the system of equations, let's go through the steps one by one.

1. **Identify which variable to substitute:**

The second equation is \(y = x + 3\). It is convenient to substitute for \(y\) in the first equation because it directly gives \(y\) in terms of \(x\).

**a)** Which variable should you substitute for in the equation \(3x + 2y = 16\)?
**Answer:** \(y\)

2. **What value can be substituted for that variable?**

From the second equation, we have \(y = x + 3\).

**b)** What value can be substituted for that variable?
**Answer:** \(x + 3\)

3. **Rewrite the equation with the substitution:**

Replace \(y\) in the first equation with \(x + 3\).

**c)** Rewrite the equation with the substitution.

Start with the original equation:
\[
3x + 2y = 16
\]
Substitute \(y\) with \(x + 3\):
\[
3x + 2(x + 3) = 16
\]

4. Now, simplify and solve for \(x\):

\[
3x + 2(x + 3) = 16
\]
Expanding this:
\[
3x + 2x + 6 = 16
\]
Combine like terms:
\[
5x + 6 = 16
\]

Next, isolate \(x\) by subtracting 6 from both sides:
\[
5x = 10
\]

Finally, divide by 5:
\[
x = 2
\]

5. Substitute \(x = 2\) back into the \(y\) equation to find \(y\):

\[
y = x + 3 = 2 + 3 = 5
\]

Thus, the solution to the system of equations is:
\[
(x, y) = (2, 5)
\]

Answer 3

Let’s go through the process step-by-step based on the given system of equations:

- Equation 1: \(3x + 2y = 16\)
- Equation 2: \(y = x + 3\)

### a) Which variable should you substitute for in the equation \(3x + 2y = 16\)?

Since Equation 2 expresses \(y\) in terms of \(x\), it is most convenient to substitute for \(y\) in Equation 1.

**Answer:** \(y\)

### b) What value can be substituted for that variable?

From Equation 2, \(y\) can be replaced with \(x + 3\).

**Answer:** \(x + 3\)

### c) Rewrite the equation with the substitution.

Now, substitute \(y\) in Equation 1 with \(x + 3\):

Starting with:
\[
3x + 2y = 16
\]
Substituting \(y\):
\[
3x + 2(x + 3) = 16
\]

### d) Solve the equation for \(x\).

Now, we will simplify and solve for \(x\):
\[
3x + 2(x + 3) = 16
\]
Expanding this:
\[
3x + 2x + 6 = 16
\]
Combine like terms:
\[
5x + 6 = 16
\]
Now, isolate \(x\) by subtracting 6 from both sides:
\[
5x = 10
\]
Finally, divide by 5:
\[
x = 2
\]

### e) Substitute that value for \(x\) into the equation and solve for \(y\).

Now substitute \(x = 2\) back into Equation 2 to find \(y\):
\[
y = x + 3 = 2 + 3 = 5
\]

### f) Use substitution to check your solution for the system.

Now we need to check if our solution \((2, 5)\) satisfies both original equations.

**Checking in Equation 1:**
\[
3x + 2y = 16
\]
Substituting \(x = 2\) and \(y = 5\):
\[
3(2) + 2(5) = 6 + 10 = 16 \quad \text{(True)}
\]

**Checking in Equation 2:**
\[
y = x + 3
\]
Substituting \(x = 2\):
\[
5 = 2 + 3 \quad \text{(True)}
\]

Since both equations are satisfied, our solution \((x, y) = (2, 5)\) is correct.

### Final Summary:
- **a:** \(y\)
- **b:** \(x + 3\)
- **c:** \(3x + 2(x + 3) = 16\)
- **d:** \(x = 2\)
- **e:** \(y = 5\)
- **f:** Both equations are satisfied, verifying the solution \((2, 5)\).