7 Point-Slope Form Homework Answers Explained

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Understanding point-slope form is a crucial aspect of algebra, allowing students to easily graph and manipulate linear equations. The point-slope form of a linear equation is given by y - y1 = m(x - x1), where m is the slope of the line and (x1, y1) is a point on the line. In this article, we will provide detailed explanations for 7 point-slope form homework answers, along with practical examples and statistical data to help solidify the concepts.

What is Point-Slope Form?

The point-slope form is a way to express a linear equation in terms of the slope and a point on the line. This form is particularly useful when you know the slope of the line and a point on the line, but not the y-intercept.

Benefits of Using Point-Slope Form

1. Easy graphing: With the point-slope form, you can easily graph a linear equation by plotting the point (x1, y1) and using the slope to find another point on the line.
2. Converting between forms: The point-slope form can be easily converted to the slope-intercept form (y = mx + b) or the standard form (Ax + By = C).
3. Solving systems of equations: The point-slope form can be used to solve systems of linear equations, especially when the equations are given in different forms.

Example 1: Writing a Linear Equation in Point-Slope Form

Find the point-slope form of the linear equation that passes through the point (2, 3) with a slope of 4.

To write the equation in point-slope form, we need to substitute the values of the point (x1, y1) = (2, 3) and the slope (m) = 4 into the equation y - y1 = m(x - x1).

y - 3 = 4(x - 2)

This is the point-slope form of the linear equation.

Example 2: Converting Point-Slope Form to Slope-Intercept Form

Convert the point-slope form equation y - 3 = 4(x - 2) to slope-intercept form.

To convert the equation, we need to isolate y by adding 3 to both sides of the equation.

y = 4(x - 2) + 3

Expanding the right-hand side of the equation, we get:

y = 4x - 8 + 3

y = 4x - 5

This is the slope-intercept form of the linear equation.

Example 3: Solving a System of Linear Equations Using Point-Slope Form

Solve the system of linear equations using the point-slope form:

y - 2 = 3(x - 1) y - 4 = 2(x - 3)

To solve the system, we need to find the intersection point of the two lines. We can do this by equating the two equations and solving for x.

3(x - 1) = 2(x - 3)

Expanding and simplifying the equation, we get:

3x - 3 = 2x - 6

Subtracting 2x from both sides of the equation, we get:

x = 3

Substituting x = 3 into one of the original equations, we get:

y - 2 = 3(3 - 1)

y - 2 = 6

y = 8

Therefore, the solution to the system is (3, 8).

Example 4: Finding the Equation of a Line Passing Through Two Points

Find the equation of the line passing through the points (2, 3) and (4, 5).

To find the equation, we need to find the slope of the line using the formula:

m = (y2 - y1) / (x2 - x1)

m = (5 - 3) / (4 - 2)

m = 2 / 2

m = 1

Now that we have the slope, we can use the point-slope form to write the equation of the line. We can use either point (2, 3) or (4, 5) as the point (x1, y1).

Using point (2, 3), we get:

y - 3 = 1(x - 2)

y - 3 = x - 2

y = x + 1

This is the equation of the line passing through the two points.

Example 5: Graphing a Linear Equation in Point-Slope Form

Graph the linear equation y - 2 = 3(x - 1) in point-slope form.

To graph the equation, we need to plot the point (x1, y1) = (1, 2) and use the slope to find another point on the line.

Using the slope, we can find another point on the line by moving 3 units up and 1 unit to the right from the point (1, 2). This gives us the point (2, 5).

Plotting the two points and drawing a line through them, we get the graph of the linear equation.

Example 6: Finding the x-Intercept of a Linear Equation in Point-Slope Form

Find the x-intercept of the linear equation y - 2 = 3(x - 1).

To find the x-intercept, we need to set y = 0 and solve for x.

0 - 2 = 3(x - 1)

-2 = 3x - 3

Adding 3 to both sides of the equation, we get:

1 = 3x

Dividing both sides of the equation by 3, we get:

x = 1/3

Therefore, the x-intercept of the linear equation is (1/3, 0).

Example 7: Finding the y-Intercept of a Linear Equation in Point-Slope Form

Find the y-intercept of the linear equation y - 2 = 3(x - 1).

To find the y-intercept, we need to set x = 0 and solve for y.

y - 2 = 3(0 - 1)

y - 2 = -3

Adding 2 to both sides of the equation, we get:

y = -1

Therefore, the y-intercept of the linear equation is (0, -1).

Now that we have worked through these examples, we hope you have a better understanding of point-slope form and how to use it to solve problems. Remember to practice, practice, practice to become more comfortable with this concept.

If you have any questions or need further clarification on any of the examples, please don't hesitate to ask. We are here to help.