Topics
2. Equations & Inequalities in Two Variables
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United States of AmericaFL
Grade 912

2.03 Point-slope and connecting forms

Lesson
Practice
Lesson

Concept summary

When we are given the coordinates of a point on the line and the slope of that line, then point-slope form can be used to state the equation of the line.

Coordinates can be given as an ordered pair, in a table of values, read from a graph, or described in a scenario. The slope can be stated as a value, calculated from two points, read from a graph, or given as a rate of change in a scenario.

Point-slope form of a linear relationship:

\displaystyle y-y_1=m\left(x-x_1\right)
\bm{m}
The slope of the line
\bm{x_1}
The x-coordinate of the given point
\bm{y_1}
The y-coordinate of the given point
Coordinate

A number used to locate a point on a number line. One of the numbers in an ordered pair, or triple, that locates a point on a coordinate plane or in coordinate space, respectively.

Ordered pair

A point on a graph written as \left(x, y\right). Also called coordinates, or a coordinate pair.

We can also find the equation in point-slope form when given the coordinates of two points on the line, by first finding the slope of the line.

We now have three forms of expressing the same equation, and each provides us with useful information about the line formed.

Point-slope form is useful when we know, or want to know:

  • The slope of the line
  • A point on the line

Slope-intercept form is useful when we know, or want to know:

  • The slope of the line
  • The y-intercept of the line

Standard form is useful when we know, or want to know:

  • Both intercepts of the line

Worked examples

Example 1

For each of the following equations, determine if they are in point-slope form, standard form, or slope-intercept form. If they are not in standard form, convert them to standard form.

a

2x+8y=10

Solution

This is in standard form as all of the variables are on one side and the constant is on the other side.

b

y=4x-10

Solution

This is in slope-intercept form.

\displaystyle y\displaystyle =\displaystyle 4x-10State the given equation
\displaystyle -4x+y\displaystyle =\displaystyle -10Subtract 4x from both sides
\displaystyle 4x-y\displaystyle =\displaystyle 10Divide both sides by -1

4x-y=10 is in standard form.

c

y-3=-\dfrac{2}{5}\left(x+7\right)

Solution

This is in point-slope form.

\displaystyle y-3\displaystyle =\displaystyle -\dfrac{2}{5}\left(x+7\right)State the given equation
\displaystyle 5y-15\displaystyle =\displaystyle -2\left(x+7\right)Multiply both sides by 5
\displaystyle 5y-15\displaystyle =\displaystyle -2x-14Distribute the multiplication
\displaystyle 2x+5y\displaystyle =\displaystyle 1Add 2x and 15 to both sides

2x+5y=1 is in standard form.

Reflection

There are more steps to convert from point-slope form to standard form, than from point-slope form to slope-intercept form.

d

x=8

Solution

Yes, this is in standard form.

Reflection

This is a vertical line.

Example 2

A line passes through the two points \left(-3,7\right) and \left(2,-3\right). Write the equation of the line in point-slope form.

Approach

We will first find the slope of the line using the two points. Then we will pick one of the points to substitute into the point-slope equation.

Solution

Find the slope of the line:

\displaystyle m\displaystyle =\displaystyle \dfrac{y_2-y_1}{x_2-x_1}Formula for slope of a line
\displaystyle =\displaystyle \dfrac{-3-7}{2-\left(-3\right)}Substitute in the points
\displaystyle =\displaystyle \dfrac{-10}{5}Simplify the subtraction
\displaystyle =\displaystyle -2Simplify the fraction

Find the equation of the line in point-slope form:

\displaystyle y-y_1\displaystyle =\displaystyle m\left(x-x_1\right)Formula for point-slope form
\displaystyle y-7\displaystyle =\displaystyle -2\left(x-\left(-3\right)\right)Substitute known values into the formula
\displaystyle y-7\displaystyle =\displaystyle -2\left(x+3\right)Simplify

Reflection

We can confirm we have correctly written this in point-slope form: y-y_1=m\left(x-x_1\right), as we can see the coordinates \left(-3,7\right), and a slope of -2 are represented correctly. This is easier to see in the previous line: y-7=-2\left(x-\left(-3\right)\right)

Example 3

A carpenter charges for a day's work using the given equation, where y is the cost and x is the number of hours worked:

\displaystyle y-125\displaystyle =\displaystyle 50\left(x-2\right)
a

Draw the graph of the linear equation from point-slope form.

Approach

We need two points to plot a line. We can read from the equation that one point on the line is \left(2,125\right). To get another point, we can use the slope from the given point or substitute in an x-value in the domain and solve for y.

Solution

1
2
3
4
x
25
50
75
100
125
150
175
200
225
y

The given point is \left(2,125\right).

Since the slope is 50, we can go up 50 units and right 1 unit from our given point to plot another point on the line.

b

The carpenter works a maximum of 10 hours per day. State the domain constraints for this scenario.

Solution

Domain: 0 \leq x \leq 10

Outcomes

MA.912.AR.1.1

Identify and interpret parts of an equation or expression that represent a quantity in terms of a mathematical or real-world context, including viewing one or more of its parts as a single entity.

MA.912.AR.2.2

Write a linear two-variable equation to represent the relationship between two quantities from a graph, a written description or a table of values within a mathematical or real-world context.

MA.912.AR.2.4

Given a table, equation or written description of a linear function, graph that function, and determine and interpret its key features.

MA.912.AR.2.5

Solve and graph mathematical and real-world problems that are modeled with linear functions. Interpret key features and determine constraints in terms of the context.

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