Question

This graph shows the lines for these equations:

3y=3x + 6

y = x + 2



How many solutions are there for this system of equations?

Answer 1

The table shows a linear relationship between the variables x and y.
x y
5 16.5
7 21.5
11 31.5
What are the slope and y-intercept of the linear relationship?

Answer 2

To determine how many solutions there are for the system of equations represented by the lines \(3y = 3x + 6\) and \(y = x + 2\), we can start by rewriting the first equation in the slope-intercept form (y = mx + b).

1. Start with the equation \(3y = 3x + 6\).
2. Divide every term by 3 to solve for \(y\):

\[
y = x + 2
\]

Now we have both equations for the lines:

1. \(y = x + 2\) (from the first equation)
2. \(y = x + 2\) (the second equation)

Both equations are the same: \(y = x + 2\).

Since these two lines are identical, they overlap completely. This means that there are infinitely many solutions to this system of equations, as every point on the line is a solution.

**Conclusion: There are infinitely many solutions.**

Answer 3

To find the slope and y-intercept of the linear relationship between \(x\) and \(y\), we can use the points provided in the table:

- (5, 16.5)
- (7, 21.5)
- (11, 31.5)

### Step 1: Calculate the slope (m)

The formula for the slope \(m\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:

\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]

Let's use the points (5, 16.5) and (7, 21.5):

1. \(x_1 = 5\), \(y_1 = 16.5\)
2. \(x_2 = 7\), \(y_2 = 21.5\)

Substituting these values into the slope formula:

\[
m = \frac{21.5 - 16.5}{7 - 5} = \frac{5}{2} = 2.5
\]

### Step 2: Calculate the y-intercept (b)

We can use the slope-point form of the equation of a line:

\[
y = mx + b
\]

We already know the slope \(m = 2.5\). We'll use one of the points to solve for \(b\). Let's use the point (5, 16.5):

\[
16.5 = 2.5(5) + b
\]

Calculating \(2.5(5)\):

\[
16.5 = 12.5 + b
\]

Now isolate \(b\):

\[
b = 16.5 - 12.5 = 4
\]

### Conclusion

The slope of the linear relationship is \(m = 2.5\) and the y-intercept is \(b = 4\).

So, the final results are:

- **Slope (m)**: 2.5
- **Y-intercept (b)**: 4

The equation of the line is:

\[
y = 2.5x + 4
\]