Writing Equations of Lines Starting with Point – Slope Form y – y1 = m (x – x1)

Note: This equation is not in function form … more on that later. Point – Slope Form To write an equation of a line in point – slope form, all you need is … … Any Point On The Line … (x1, y1) … The Slope … m Once you have these two things, you can write the equation as y – y1 = m (x – x1) That’s “y minus the y-value of the point equals the slope times the quantity of x minus the x-value of the point”. Note: This equation is not in function form … more on that later. DONE

Example #1 Write the equation of the line that goes through the point (2, –3) and has a slope of 4. Point = (2, –3) Slope = 4 Starting with the point – slope form y – y1 = m (x – x1) Plug in the y-value, the slope, and the x-value to get y + 3 = 4 (x – 2) Notice, that when you subtracted the “–3” it became “+3”. DONE

Example #2 Write the equation of the line that goes through the point (–4, 6) and has a slope of . Point = (–4, 6) Slope = Starting with the point – slope form y – y1 = m (x – x1) Plug in the y-value, the slope, and the x-value to get y – 6 = (x + 4) Notice, that when you subtracted the “–4” it became “+4”. DONE

Using the first point, we have, Example #3 Write the equation of the line that goes through the points (6, –4) and (2, 8) . We have two points, but we’re missing the slope. Using the formula for slope, we can find the slope to be y2 – y1 x2 – x1 To use point – slope form, we need a point and a slope. Since we have two points, just pick one … IT DOESN’T MATTER … BOTH answers are acceptable… more on why later. Using the first point, we have, Using the second point, we have, Point = (6, –4) Slope = –3 Point = (2, 8) Slope = –3 y + 4 = –3 (x – 6) y – 8 = –3 (x – 2) DONE

Other Forms of Linear Equations So far, we have discussed only point-slope form. There are other forms of equations that you should be able to identify as a line and graph if necessary. Horizontal Line: y = c , where c is a constant. Example: y = 3 Vertical Line: x = c , where c is a constant. Example: x = –6 Slope – Intercept Form: y = mx + b To write equations in the last two forms, start in point – slope form and rearrange the variables to match the correct format. The next few slides will cover how to do this. m = the slope of the line … b = the y-intercept Example: y = 3x – 6 Standard Form: Ax + By = C A, B, and C are integers. Example: 3x + 4y = –36 DONE

Writing Equations in Slope – Intercept Form Earlier (click here to review) we wrote an equation of the line that went through the points (6, –4) and (2, 8) . Sometimes, we want the line written in a different form. To change a point-slope equation in slope-intercept form, solve for y and simplify the right side of the equation. - Solve for y: Add or subtract the y-value of the point to both sides - Simplify: Distribute the slope and then combine like terms. Here are the two answers we had from the earlier example. y + 4 = –3 (x – 6) y – 8 = –3 (x – 2) Subtract 4 from both sides SOLVE FOR y Add 8 to both sides y = –3 (x – 6) – 4 y = –3 (x – 2) + 8 Distribute –3 and combine like terms SIMPLIFY Distribute –3 and combine like terms y = –3x + 18 – 4 y = –3x + 14 y = –3x + 6 + 8 y = –3x + 14 Notice … They’re the same! DONE

Example #4 Write the equation of the line in slope-intercept form that goes through the point (6, 2) and has slope . Begin in point-slope form: y – 2 = (x – 6) Add 2 to both sides y = (x – 6) + 2 Solve for y: Distribute: y = x – 4 + 2 Combine Like Terms: y = x – 2 DONE

Writing Equations In Standard Form The last form of a linear equation we are going to cover is called Standard Form. Ax + By = C , where A, B, and C are integers. If you needed to write an equation of a line in standard form, you would start in point-slope form or slope-intercept form, depending on what information you are given. In both cases, you must put all variables on the left side and all constant values on the right side. If any of the coefficients (A, B, or C) are NOT integers, then you must eliminate any fractions or decimals by multiplying every term in the equation by the appropriate factor. DONE

Example #5 Rewrite each of the equations below in standard form. y = x – 4 y – 6 = (x + 4) Distribute Subtract from both sides. Subtract from both sides, and add 6 to both sides. Multiply ALL terms by 3 in order to eliminate the fraction. –2x + 3y = –12 Multiply ALL terms by 2. – 3x + 2y = 24 DONE

y as a FUNCTION of x For an equation to be written as a function, you must solve for y. Solving for y means that “y is written as a function of x ”. Of the three types of linear equations discussed in this presentation, only slope-intercept form is written as a function. When your equation is in point – slope form simply add or subtract the y-value of the point to the other side. From our first example we had y + 3 = 4 (x – 2) In order to write y as a function of x we subtract 3 from both sides of the equation. y = 4 (x – 2) – 3 When you write y as a function of x, you have put your equation in function form. You may replace the y with the notation f (x) … read “f of x ” or “function of x ”. f (x) = 4 (x – 2) – 3 DONE