1 Chapter 2, Section 4 Writing Linear Equations. 2 Equations of Lines Remember that we know how to write lines in STANDARD FORM: Ax + By = C. Another.

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1 Chapter 2, Section 4 Writing Linear Equations

2 Equations of Lines Remember that we know how to write lines in STANDARD FORM: Ax + By = C. Another form we should be familiar with is the SLOPE-INTERCEPT form (because it tells us the slope of the line and the y- intercept): y = mx + b, where m is the slope and b is the y-intercept.

3 The NEW (old?) Equation: Point- Slope form We call it the point slope form because … it tells us a point on the line and the line’s slope. More useful, however, is that if we know a point on the line and the slope of the line, we can use it to write the equation of that line. Is it NEW? No, it comes to us from the equation for slope: Formula for slope Multiply both sides by x 2 - x 1 Symmetric Property of Equality We won’t concern ourselves with Point 2; we only need ONE point.

4 Example 1 Write an equation in slope-intercept form for the line that has a slope of -(3/5) and passes through (5,-2). Since we are given ONE point (point one!) and a slope, let us use the point-slope form: y - y 1 = m(x - x 1 )Point-slope form y - (-2) = -(3/5)(x - 5)Substitution y + 2 = -(3/5)x + 3Simplify and distribute y = -(3/5)x + 1Solve for y -- slope- intercept form

5 Example 2 What is the equation of the line through (2,-3) and (-3,7)? a)y = -2x - 1c)y = (1/2)x + 1 b)y = -(1/2)x + 1d)y = -2x + 1 Given two points we can find the slope, and using the slope and ONE of the points, we can write an equation to turn into slope intercept form. First find the slope: Plug the slope and EITHER point into the point slope form and convert to slope-intercept form: So, D is the correct answer!

6 Example 3 Write the equation for the line that passes through (3,- 2) and is perpendicular to the line whose equation is y = -5x + 1. First, we see that the slope of the line we want to be perpendicular to is -5, so we know the slope of a line perpendicular is… … 1 / 5. We have a point on the line, (3,-2) and the slope of the line perpendicular: 1 / 5. Point-slope form to the rescue! y - y 1 = m(x - x 1 )Point-slope form y - (-2)= 1 / 5 (x - 3)Substitute y + 2 = 1 / 5 x - 3 / 5 Simplify y = 1 / 5 x - 13 / 5 Slope-intercept form

7 Example 4: Real-world application Jean is trying to figure out her pay, which is part salary and part commission. When her sales are $100, she makes $58. When her sales are $300, she makes $78. What is her daily salary and her commission percentage? We recognize the situation to be a slope-intercept form: Pay = percentage of sales+ salary y = m · x+ b In this case, her pay is a FUNCTION of her sales -- her pay changes only as her sales change: y = f(x). We will then do what we did in the last example: Find the slope -- her percentage commission, then put her slope and either point (100,58) or (300,78) into the point- slope form to get the salary.

8 Example 4 (continued) Finding the slope, or commission: Finding the y-intercept, or salary: So her commission is 10% ( 1 / 10 ) and her daily salary is $48.