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PreCalculus

PreCalculus. 1.2 – Linear Equations in Two Variables. Using Slope. The simplest mathematical model for relating two variables is the linear equation in two variables y = mx + b (slope intercept form), where m represents the slope and b represents the y-intercept.

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PreCalculus

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  1. PreCalculus 1.2 – Linear Equations in Two Variables

  2. Using Slope • The simplest mathematical model for relating two variables is the linear equation in two variables y = mx + b (slope intercept form), where m represents the slope and b represents the y-intercept. The slope of a nonvertical line is the number of units the line rises (or falls) vertically for each unit of horizontal change from left to right

  3. Example 1 – Sketching Lines Sketch the lines through the point with the indicated slopes. • Point: (3, 2) Slope: ½ • Point: (-1, 3) Slope: -2

  4. Example 2 – Graphing a Linear Equation Sketch the graph of each linear equation: A) y = 2x + 1 B) y = 2 • x + 2y = 4 D) x + 2 = -1

  5. Example 3 – Using Slope as a Ratio The maximum recommended slope of a wheelchair ramp is 1/12. A business is installing a wheelchair that rises 22 inches over a horizontal length of 24 feet. Is the ramp steeper than recommended?

  6. Example 4 – Using Slope as a Rate of Change A kitchen appliance manufacturing company determines that the total cost in dollars of producing x units of a blender is C = 25x + 3500 (cost equation). Describe the practical significance of the y-intercept and slope of this line.

  7. Finding the Slope of a Line Slope = Slope = Slope =

  8. Example 5 – Finding the Slope of a Line Through Two Points Find the slope of the line passing through each pair of points: • (-2, 0) and (3, 1) (b) (-1, 2) and (2, 2) • (0, 4) and (1, -1) (d) (3, 4) and (3, 1)

  9. Example 6 – Using the Slope-Intecept Form Find the slope-intercept form of the equation of the line that has a slope of 3 and passes through the point (1, -2).

  10. Example 7 – Equation of a Line through Two Points Find the slope-intercept form of the equation of the line passing through the points (2, 4) & (-1, 5).

  11. Example 8 – Predicting Revenue per Share The revenue per share for eBay Inc. was $0.20 in 1998 and $0.91 in 1999. Using only this information, write a linear equation that gives the revenue per share in terms of the year. Then predict the revenue per share for 2000.

  12. Extrapolation • The prediction method in example 8 is called linear extrapolation. Extrapolation occurs when a predicted point is outside the range of the two given points.

  13. Interpolation • Interpolation occurs when a predicted point is inside the range of the two given points.

  14. Equations of Lines General Form: Ax + By + C = 0 Vertical Line x = a (where a is the x-intercept) Horizontal line: x = b Slope-Intercept Form: y = mx + b

  15. Parallel & Perpendicular Lines • Two distinct non-vertical lines are parallel if and only if their slopes are equal. Example: y = 2x + 3 & y = 2x – 4 are parallel lines • Two non-vertical lines are perpendicular if and only if their slopes are negative reciprocals of each other. • Example: y = -½x – 1 & y = 2x + 3 are perpendicular lines

  16. Example 9 – Finding Parallel and Perpendicular Lines Find the slope-intercept forms of the equation of the lines that pass through the point (2, -1) and are (a) parallel to and (b) perpendicular to the line 2x – 3y = 6.

  17. Example 10 – Straight Line Depreciation Your publishing company has purchased a $12,000 machine that has a useful life of 8 years. The salvage value at the end of 8 years is $2000. Write a linear equation that describes the book value of the machine each year.

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