OPENER/REVIEW: Write the linear equation through the points (0,5) and (3,7). Then, sketch the graph.

To find the linear equation passing through the points \( (0, 5) \) and \( (3, 7) \), we can use the **slope-intercept form** of a line equation, which is:

\[
y = mx + b
\]

### Step 1: Find the slope (\(m\))
The formula for the slope between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is:

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

Substituting the points \( (0, 5) \) and \( (3, 7) \):

\[
m = \frac{7 - 5}{3 - 0} = \frac{2}{3}
\]

So, the slope \(m = \frac{2}{3}\).

### Step 2: Find the y-intercept (\(b\))
Since we know the slope \(m = \frac{2}{3}\) and one of the points is \( (0, 5) \), we can substitute into the equation \( y = mx + b \). Using \( (0, 5) \), we have:

\[
5 = \frac{2}{3}(0) + b
\]

\[
b = 5
\]

### Step 3: Write the equation
Now that we have both the slope and the y-intercept, the equation of the line is:

\[
y = \frac{2}{3}x + 5
\]

### Graph Sketching
To sketch the graph of this line:

1. Plot the y-intercept \( (0, 5) \).
2. Use the slope \( \frac{2}{3} \) to find another point. From \( (0, 5) \), move up 2 units and to the right 3 units to reach the point \( (3, 7) \).
3. Draw a straight line passing through these points.

If you'd like, I can generate a graph for you.

Do you want further details or have any questions?

### Related Questions:
1. How do you calculate the slope between two points?
2. What is the significance of the y-intercept in a linear equation?
3. How would the equation change if the slope were negative?
4. What are some real-life examples of linear relationships?
5. How can you find the x-intercept of a linear equation?

### Tip:
When graphing a line, always start by plotting the y-intercept, as it gives a fixed point from which to measure the slope.

Write the linear equation through the points (0,5) and (3,7). Then, sketch the graph.

Learn how to write the linear equation through the points (0, 5) and (3, 7) using the slope-intercept form. This step-by-step solution covers key algebra concepts, including the slope formula and how to find the y-intercept. Suitable for students in Grades 6-8.

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