Linear Relations and Functions B4 Writing Linear Equations

Linear Relations and Functions B-4 Writing Linear Equations

Warm – up Connecting Prior Knowledge

Objectives Write an equation of a line given the slope and a point on the line Write an equation of a line parallel or perpendicular to a given line

Essential Question How do you write an equation of a line given the coordinates of two points on a line?

Slope-Intercept Form of a Linear Equation The slope-intercept form of the equation of a line is y = mx + b, where m is the slope and b is the y-intercept. y = mx + b slope y-intercept

If you are given the slope and y-intercept of a line, you can quickly find an equation of the line by substituting the values of m and b into the slope-intercept form. For example, if you know that the slope of a line is – 3 and the y -intercept is 4, the equation of the line is y = – 3 x + 4, or, in standard form, 3 x + y = 4. In previous courses you have also learned that the slope-intercept form can be used to find an equation of a line given the slope and the coordinates of any point on the line.

Write an equation in slope-intercept form for the line that has a slope of and passes through (5, – 2). Slope-intercept form – Simplify. Add 3 to each side. Answer: The y-intercept is 1. So, the equation in slope-intercept form is

More often in algebra, when you are given slope and a point or two points on a line most students find it difficult and tedious to solve for the y-intercept (b) in the slope-intercept form. An easier method to use given slope and a point or given two coordinates on a line is the point-slope form.

Point-Slope Form of a Linear Equation The point-slope form of the equation of a line is y – y 1 = m(x – x 1), where (x 1, y 1) are the coordinates of a point on the line and m is the slope of the line. y – y 1 = m(x – x 1) slope coordinates of point on line

Write an equation in slope-intercept form for the line that has a slope of and passes through (– 3, – 1). Answer:

Multiple-Choice Test Item What is an equation of the line through (2, – 3) and (– 3, 7)? A B C D Read the Test Item You are given the coordinates of two points on the line. Notice that the answer choices are in slope-intercept form.

Solve the Test Item First, find the slope of the line. Slope formula Simplify. The slope is – 2. That eliminates choices B) and C).

Then use the point-slope formula to find an equation. Point-slope form you can use either point for. Distributive Property Subtract 3 from each side. Answer: D

Multiple-Choice Test Item What is an equation of the line through (2, 5) and (– 1, 3)? A C B D Answer: C

When changes in real-world situations occur at a linear rate, a linear equation can be used as a model for describing the situation. Sales As a part-time salesperson, Jean Stock is paid a daily salary plus commission. When her sales are $100, she makes $58. When her sales are $300, she makes $78. Write a linear equation to model this situation. Let x be her sales and let y be the amount of money she makes. Use the points (100, 58) and (300, 78) to make a graph to represent the situation.

Slope formula Simplify.

Now use the slope and either of the given points with the point-slope form to write the equation. Point-slope form Distributive Property Add 58 to each side. Answer: is The slope-intercept form of the equation

What are Ms. Stock’s daily salary and commission rate? The y-intercept of the line is 48. The y-intercept represents the money Jean would make if she had no sales. Thus, $48 is her daily salary. The slope of the line is 0. 1. Since the slope is the coefficient of x, which is her sales, she makes 10% commission. Answer: Ms. Stock’s daily salary is $48, and she makes a 10% commission.

How much would Jean make in a day if her sales were $500? Find the value of y when Use the equation you found in Example 3 a. Replace x with 500. Simplify. Answer: She would make $98 if her sales were $500.

Sales The student council is selling coupon books to raise money for the Humane Society. If the group sells 200 books, they will receive $150 dollars. If they sell 500 books, they will make $375. a. Write a linear equation to model this situation. Answer: b. Find the percentage of the proceeds that the student council Answer: 75%receives. c. If they sold 1000 books, how much money would they receive to donate to the Humane Society? Answer: $750

The slope-intercept and point-slope forms can be used to find equations of lines that are parallel or perpendicular. Write an equation for the line that passes through (3, – 2) and is perpendicular to the line whose equation is The slope of the given line is – 5. Since the slopes of perpendicular lines are opposite reciprocals, the slope of the perpendicular line is Use the point-slope form and the ordered pair (3, – 2) to write the equation.

Point-slope form Distributive Property Add – 2 to each side. Answer: An equation of the line is

Write an equation for the line that passes through (3, 5) and is perpendicular to the line whose equation is Answer:

Essential Question How do you write an equation of a line given the coordinates of two points on a line? Find the slope using the slope formula and use the pointslope form, y – y 1 = m(x – x 1), to substitute the slope for m and one of the given points for (x 1, y 1).

Math Humor Student: I’ll just draw a quick line by hand guess the slope. Teacher: No, no. It’s point-slope form, not point-sloppy form.