Writing Linear Equations fun with two variables

Writing Linear Equations Overview Point-Slope Form: y – y1 = m(x – x1) Slope-Intercept Form: y = mx+b Standard Form: Ax + By = C Parallel Lines: Slopes Equal Perpendicular Lines: Slopes Opposite Reciprocals

#1 – Calculating Slope w/ 2 Points Calculate the slope of a line passing through the points (–7, 9) and (2, 3). Recall the slope formula: Assign points and substitute into slope formula m = (3 – 9)/(2 – (–7)) Substitute m = –6/9 Evaluate m = –2/3 Simplify

#1 – Calculating Slope w/ 2 Points Special Cases: (–2, 1), (–2, –4) (5, 1), (–3, 1)

#1 – Practice Calculate m (slope) between each set of points. (2, 4), (5, 13) m = 9/3 or 3 (–1, 2), (4, –1) m = –3/5 (–6, 5), (–6, –3) m = undefined

Warmup – Identify Components Identify m (slope) and y-intercept. y = 2/3x – 13 Identify (x1, y1) and m. y + 2 = –1/5(x + 5)

#2 – Point-Slope w/ 2 Points Write a linear equation in point-slope form of the line passing through the points (0, 4) and (1, 6). 1st Step – Find the slope between the two points. m = 6 – 4/1 – 0 = 2 2nd Step – Substitute slope and any known point into point-slope equation. y – y1 = m(x – x1) y – (6) = 2(x – (1)) Simplify and rewrite final linear equation y – 6 = 2(x – 1) OR y – 4 = 2(x – 0)

#2 – Point-Slope from Graph Write a linear equation in point-slope form of line shown. 1st Step – Identify slope of line m = –1 – (–5)/3 – 0 = 4/3 2nd Step – Substitute slope and any known point into point-slope equation. y – y1 = m(x – x1) y – (–1) = 4/3(x – (3)) Simplify and rewrite final linear equation y + 1 = 4/3(x – 3) OR y + 5 = 4/3(x – 0)

#2 – Point-Slope w/ Table of Values Write a linear equation in point-slope form of the line passing through the points in the table. 1st Step – Identify slope of line with any two points m = 0 – 3/8 – 4 = –3/4 2nd Step – Substitute slope and any known point into point-slope equation y – y1 = m(x – x1) y – (12) = –3/4(x – (–8)) Simplify and rewrite final linear equation y – 12 = –3/4(x + 8) x y -8 12 -4 9 4 3 8

Warmup – Linear vs. Proportional After 3 months of gym membership, Mrs. Alves had paid $110. After 7 months, she had paid $190.

Warmup – Problem Solving Mr. Thomas bought his first cell phone. His service is $45 per month. After 7 months he paid $464. How much did he pay for the actual cell phone? 1st Step – Substitute slope into equation. y – y1 = 45(x – x1) 2nd Step – Substitute known coordinate pair into equation. y – 464 = 45(x – 7) Simplify and rewrite to slope-intercept form. y – 464 = 45x – 315 y = 45x + 149 The y-intercept equals 149, therefore the phone cost $149.

#3 – Rewrite Equations (Literal Equations) Literal Equation – an equation expressed in terms of variables, symbols, and constants. To solve a literal equation for any particular unknown, determine the order of PEMDAS performed on that unknown, then REVERSE and INVERSE. If unknown is within parentheses, try to factor using the distributive property If dealing with exponents or radicals, use the inverse to raise exponent to power of 1 Complete when unknown is isolated on one side of equation

#3 – Rewrite Equations (Literal Equations) Rewrite the equation below to isolate m Rewrite the equation below to isolate t

#3 – Rewrite Equations (Literal Equations) Rewrite the equation below to isolate x

#4 – Slope-Intercept w/ 2 Points Write a linear equation in slope-intercept form of the line passing through the points (0, 4) and (1, 6). 1st Step – Find the slope between the two points. m = 6 – 4/1 – 0 = 2 2nd Step – Write equation to point-slope form. y – y1 = m(x – x1) y – 6 = 2(x – 1) OR y – 4 = 2(x – 0) Simplify and rewrite final linear equation in slope-intercept y = 2x + 4

Warmup – Linear vs. Proportional Little Timmy is 5 years old and 35 inches tall. If he continues to grow at the same rate, how tall will he be when he is 10 years old?

#4 – Practice Write a linear equation in slope-intercept form: (2, 5), (0, 13) y = –4x + 13 (–2, 2), (6, –4) y = –3/4x + 1/2 (–6, –12), (–8, –17) y = 5/2x + 3

Warmup – Problem Solving After 5 months of having a cell phone, Mr. Thomas paid a total of $314. After 12 months, he paid a total of $685. Write a linear equation to model the total amount he has paid as a function of months of service.

Warmup – Write an Equation Write a linear equation of the line passing through the points (2, 7) and (-1, 8) in BOTH: Point-Slope Form y – 7 = –1/3(x – 2) OR y – 8 = –1/3(x + 1) Slope-Intercept Form y = –1/3x + 23/3 OR y = –1/3x + 7 2/3

What is Standard Form? Ax + By = C, where: A, B, and C are integers Fractional or decimal coefficients must be removed by multiplication Can be multiplied or divided by any number, given that operation is performed on both sides of equation Can be obtained by rewriting either point-slope or slope-intercept form equations A and B CAN’T BOTH equal 0 (one or the other can equal 0)

#5 – Rewriting to Standard Form Rewrite the point-slope equation to standard form: y – 8 = 2/3(x + 3) Distribute: y – 8 = 2/3x + 2 Move x term: –2/3x + y – 8 = 2 Move constant: –2/3x + y = 10 Multiply common multiple: 3(–2/3x + y) = 3(10) Simplify and rewrite: –2x + 3y = 30 Check of Equivalence using (0, 10) (10) – 8 = 2/3((0) + 3) –2(0) + 3(10) = 30 2 = 2 30 = 30

#5 – Rewriting to Standard Form Rewrite the slope-intercept equation to standard form: y = 2/7x – 14 Move x term: –2/7x + y = –14 Multiply common multiple: 7(–2/7x + y) = 7(–14) Simplify and rewrite: –2x + 7y = –98 Check of Equivalence using (0, –14) (–14) = 2/7(0) – 14 –2(0) + 7(–14) = –98 –14 = –14 –98 = –98

#6 – Equivalent Standard Form Equations Rewrite an equivalent standard form equation to: 2x + 3y = 12 Multiply both sides by 4: 4(2x + 3y) = 4(12) Simplify and Rewrite: 8x + 12y = 48 Check of Equivalence using (0, 4) 2(0) + 3(4) = 12 8(0) + 12(4) = 48 12 = 12 48 = 48

#6 – Equivalent Standard Form Equations Rewrite an equivalent standard form equation to: 1/5x + 2/3y = 5 Multiply both sides by 15: 15(1/5x + 2/3y) = 15(5) Simplify and Rewrite: 3x + 10y = 75 Check of Equivalence using (5, 6) 1/5(5) + 2/3(6) = 5 3(5) + 10(6) = 75 5 = 5 75 = 75

Warmup – Identify ∥ and ⊥ Slopes y + 6 = 2/3(x – 5) ∥ = 2/3 ⊥ = –3/2 y = –7x + 4 ∥ = –7 ⊥ = 1/7 3x + 5y = 18 ∥ = –3/5 ⊥ = 5/3

Review: Parallel and Perpendicular Slopes Parallel Lines Have equal slopes Example: y = 3x + 6 would be parallel to y = 3x – 9 Perpendicular Lines Have opposite reciprocal slopes (opposite sign and flipped) Example: y = –1/5x + 2 would be perpendicular to y = 5x + 6

#7 – Write Parallel Equation Write an equation of a line that passes through the point (–7, 2) and is parallel to 4x – 2y = –10. Identify key info: ∥m = 2, (x1, y1) = (–7, 2) Substitute: y – 2 = 2(x + 7) IF REQUIRED, REWRITE TO SLOPE-INTERCEPT Distribute: y – 2 = 2x + 14 Move constant: y = 2x + 16 Check using original point (–7, 2) (2) = 2(–7) + 16 2 = 2

#8 – Write Perpendicular Equation Write an equation of a line that passes through the point (9, –4) and is perpendicular to y = 3x – 6. Identify key info: ⊥m = –1/3, (x1, y1) = (9,–4) Substitute: y + 4 = –1/3(x – 9) IF REQUIRED, REWRITE TO SLOPE-INTERCEPT Distribute: y + 4 = –1/3x + 3 Move constant: y = –1/3x – 1 Check using original point (9, –4) (–4) = –1/3(9) – 1 –4 = –4

Warmup – Write Equation in 3 Forms Write the equation of the line passing through points (-4, 1) and (-2, 5) in: Point-Slope Form: y – 5 = 2(x + 2) , or y – 1 = 2(x + 4) Slope-Intercept Form: y = 2x + 9 Standard Form: -2x + y = 9

Warmup – Rewrite to Standard Form y + 4 = –3(x + 4) 3x + y = –16 y = –3/5x – 4 3x + 5y = –20

Warmup – Evaluate for x = 8 y = -4x + 7 y = –25 y + 2 = 1/4(x – 4) y = –1 5x – 2y = 54 y = –7