# How to Write Slope-intercept Form and Point-slope Form?

Slope-intercept form and point-slope form are two ways of representing a linear equation in two variables (\(x\) and \(y\)). Both of these forms are commonly used in mathematics and are useful in different contexts.

## A Step-by-step Guide to Write Slope-intercept Form and Point-slope Form

Here are the steps to write slope-intercept form and point-slope form

### Slope-Intercept Form:

#### Step 1: Write the equation

Write the equation in the form \(y = mx + b\), where \(m\) is the slope of the line and \(b\) is the \(y\)-intercept.

#### Step 2: Determine the slope of the line

The slope, represented by \(m\), is the rate at which the \(y\)-coordinate changes for every unit change in the \(x\)-coordinate. You can determine the slope by looking at the coefficient of \(x\) in the equation.

#### Step 3: Determine the \(y\)-intercept of the line

The \(y\)-intercept, represented by \(b\), is the point where the line crosses the \(y\)-axis. You can determine the \(y\)-intercept by looking at the constant term in the equation.

#### Step 4: Plot the y-intercept on the y-axis

This is the point where the line intersects the \(y\)-axis.

#### Step 5: Use the slope to determine additional points on the line

To do this, you can start from the \(y\)-intercept and move in the direction of the slope. For example, if the slope is positive, move up and to the right; if the slope is negative, move down and to the right.

#### Step 6: Plot the additional points on the line

Plot the additional points on the line and draw a straight line through all the points. This is the graph of the line represented by the equation in slope-intercept form.

### Point-Slope Form:

#### Step 1: Write the equation

Write the equation in the form \(y – y_1 = m(x – x_1)\), where \(m\) is the slope of the line and \((x_1, y_1)\) is a point on the line.

#### Step 2: Determine the slope of the line

The slope, represented by \(m\), is the rate at which the \(y\)-coordinate changes for every unit change in the \(x\)-coordinate. You can determine the slope by looking at the coefficient of \(x\) in the equation.

#### Step 3: Determine the coordinates of the point on the line

The point on the line, represented by \((x_1, y_1)\), is a known point on the line. You can determine the coordinates of the point by looking at the values given in the equation.

#### Step 4: Plot the known point on the line

This is a point that you know is on the line represented by the equation in point-slope form.

#### Step 5: Use the slope to determine additional points on the line

To do this, you can start from the known point and move in the direction of the slope. For example, if the slope is positive, move up and to the right; if the slope is negative, move down and to the right.

#### Step 6: Plot the additional points on the line

Plot the additional points on the line and draw a straight line through all the points. This is the graph of the line represented by the equation in point-slope form.

Overall, both slope-intercept form and point-slope form provide different ways of representing linear equations in two variables and can be useful in different contexts.

### Writing Slope-intercept Form and Point-slope Form – Examples 1

Find the equation of a line passing through the point \((-2, 7)\) with a slope of \(4\), and write it in both slope-intercept and point-slope forms.

#### Solution:

To find the equation of a line passing through the point \((-2, 7)\) with a slope of \(4\):

**Slope-Intercept Form:**

The slope-intercept form of a linear equation is \(y=mx + b\), where \(m\) is the slope and \(b\) is the \(y\)-intercept. We have the slope \((m)\) as \(4\) and the point \((-2, 7)\) lies on the line. So, we can use the point-slope form to find the equation of the line.

**Point-Slope Form:**

The point-slope form of a linear equation is \(y – y_1 = m(x – x_1)\), where \((x_1, y_1)\) is a point on the line and m is the slope. Substituting the given values, we have:

\(y – 7 = 4(x – (-2))\) [Using point-slope form]

\(y – 7 = 4(x + 2)\)

\(y – 7 = 4x + 8\)

\(y = 4x + 15\) [Using slope-intercept form]

Therefore, the equation of the line passing through the point \((-2, 7)\) with a slope of \(4\) is \(y = 4x + 15\) in slope-intercept form and \(y – 7 = 4(x + 2)\) in point-slope form.

### Writing Slope-intercept Form and Point-slope Form – Examples 2

Given that a line passes through the point \((3, 2)\) and has a slope of \(-2\), find its equation in both slope-intercept and point-slope forms.

#### Solution:

To find the equation of a line passing through the point \((3, 2)\) with a slope of \(-2\):

**Slope-Intercept Form:**

We can use the slope-intercept form of a linear equation to find the equation of the line. The slope-intercept form of a linear equation is \(y=mx + b\), where \(m\) is the slope and \(b\) is the \(y\)-intercept. We have the slope \((m)\) as \(-2\) and the point \((3, 2)\) lies on the line. So, we can use the point-slope form to find the equation of the line.

**Point-Slope Form:**

The point-slope form of a linear equation is \(y – y_1 = m(x – x_1)\), where \((x_1, y_1)\) is a point on the line and m is the slope. Substituting the given values, we have:

\(y – 2 = -2(x – 3)\) [Using point-slope form]

\(y – 2 = -2x + 6\)

\(y = -2x + 8\) [Using slope-intercept form]

Therefore, the equation of the line passing through the point \((3, 2)\) with a slope of \(-2\) is \(y = -2x + 8\) in slope-intercept form and \(y – 2 = -2(x – 3)\) in point-slope form.

**Exercises for** **Writing Slope-intercept Form and Point-slope Form**

- Write the slope-intercept form of the equation of a line passing through the point \((2, 4)\) with a slope of \(1\).
- Write the point-slope form of the equation of a line passing through the point \((-1, 4)\) and \((-4,2)\).
- Write the point-slope form of the equation of a line passing through the point \((4, -4)\) with a slope of \(-2\).

- \(\color{blue}{y=x+2}\)
- \(\color{blue}{y\:-\:4=\left(\frac{2}{3}\right)\left(x+1\right)}\)
- \(\color{blue}{y\:+\:4=-2\left(x-4\right)}\)

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