Question

Graph the line that passes through the points left parenthesis, minus, 2, comma, 5, right parenthesis(−2,5) and left parenthesis, minus, 4, comma, 3, right parenthesis(−4,3) and determine the equation of the line

Answer 1

To find the equation of the line that passes through the points \((-2, 5)\) and \((-4, 3)\), we can follow these steps:

### Step 1: Find the Slope

The slope \(m\) of a line through two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the formula:

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

Here, we have:
- \((x_1, y_1) = (-2, 5)\)
- \((x_2, y_2) = (-4, 3)\)

Substituting the values in:

\[
m = \frac{3 - 5}{-4 - (-2)} = \frac{-2}{-4 + 2} = \frac{-2}{-2} = 1
\]

So, the slope \(m = 1\).

### Step 2: Use Point-Slope Form to Find the Equation

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

\[
y - y_1 = m(x - x_1)
\]

Using point \((-2, 5)\) and the slope \(m = 1\):

\[
y - 5 = 1(x - (-2))
\]

This simplifies to:

\[
y - 5 = 1(x + 2)
\]

Distributing the slope:

\[
y - 5 = x + 2
\]

### Step 3: Solve for \(y\) to Get the Slope-Intercept Form

Now, add 5 to both sides to solve for \(y\):

\[
y = x + 2 + 5
\]
\[
y = x + 7
\]

### Conclusion

The equation of the line that passes through the points \((-2, 5)\) and \((-4, 3)\) is:

\[
y = x + 7
\]

### Graphing the Line

To plot the line, you can choose several \(x\) values and calculate \(y\):

- For \(x = -2\): \(y = -2 + 7 = 5\) (Point \((-2, 5)\))
- For \(x = -4\): \(y = -4 + 7 = 3\) (Point \((-4, 3)\))
- For \(x = 0\): \(y = 0 + 7 = 7\) (Point \((0, 7)\))

Plot these points on a graph and draw a line through them. The line will have a slope of 1, meaning it rises one unit for every one unit it moves to the right.