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Math 11 Academic Linear Relations Review

Math 11 Academic Linear Relations Review. Identifying a Linear Equation. The exponent of each variable is 1. The variables are added or subtracted. The slope-intercept form of a linear equation is: y = mx + b, where m represents the slope and b represents the y-intercept

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Math 11 Academic Linear Relations Review

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  1. Math 11 AcademicLinear Relations Review

  2. Identifying a Linear Equation • The exponent of each variable is 1. • The variables are added or subtracted. • The slope-intercept form of a linear equation is: y = mx + b, where m represents the slope and b represents the y-intercept • The standard form of a linear equation is: Ax + By = C whereAorB can equal zero and A > 0 • There are no radicals in the equation. • Every linear equation graphs as a line.

  3. Examples of linear equations Equation is in Ax + By =C form Rewrite with both variables on left side … x + 6y =3 There is no y-intercept for this equation…why??? Multiply both sides of the equation by -1 … 2a – b = -5 Multiply both sides of the equation by 3 … 4x –y =-21 2x + 4y =8 6y = 3 – x x = 1 -2a + b = 5

  4. Examples of Nonlinear Equations The following equations are NOT in the standard form of Ax + By =C, or slope-intercept form ofy = mx + b: 4x2 + y = 5 xy + x = 5 s/r + r = 3 The exponent is 2 There is a radical in the equation Variables are multiplied Variables are divided

  5. x and y -intercepts • The x-intercept is the point where a line crosses the x-axis. The general form of the x-intercept is (x, 0). The y-coordinate will always be zero. • The y-intercept is the point where a line crosses the y-axis. The general form of the y-intercept is (0, y). The x-coordinate will always be zero.

  6. Finding the x-intercept • For the equation 2x + y = 6, we know that y must equal 0. What must x equal? • Plug in 0 for y and simplify. 2x + 0 = 6 2x = 6 x = 3 • So (3, 0) is the x-intercept of the line.

  7. Finding the y-intercept • For the equation 2x + y = 6, we know that x must equal 0. What must y equal? • Plug in 0 for x and simplify. 2(0) + y = 6 0 + y = 6 y = 6 • So (0, 6) is the y-intercept of the line.

  8. x-intercept: Plug in y = 0 x = 4y - 5 x = 4(0) - 5 x = 0 - 5 x = -5 (-5, 0) is the x-intercept y-intercept: Plug in x = 0 x = 4y - 5 0 = 4y - 5 5 = 4y = y (0, ) is the y-intercept Find the x and y- interceptsof x = 4y – 5

  9. x y Find the x and y-intercepts of x = 3 • x-intercept • Plug in y = 0. There is no y. Why? • x = 3 is a verticalline so x always equals 3. • (3, 0) is the x-intercept. • y-intercept • A vertical line never crosses the y-axis. • There is no y-intercept.

  10. Vertical Lines • Slope is undefined. • Equation form is x = #. • Example 1: • Write an equation of a line and graph it with undefined slope and passes through (1, 0). • x = 1 • Example 2: • Write an equation of a line that passes through (3, 5) and (3, -2). • x = 3

  11. x y Find the x and y-intercepts of y = -2 • y-intercept • y = -2 is a horizontal line so y always equals -2. • (0,-2) is the y-intercept. • x-intercept • Plug in y = 0. y cannot = 0 because y = -2. • y = -2 is a horizontal line so it never crosses the x-axis. • There is no x-intercept.

  12. Horizontal Lines • Slope is zero. • Equation form is y = #. • Example 1: • Write an equation of a line and graph it with zero slope and y-intercept of -2. • y = -2 • Example 2: • Write an equation of a line and graph it that passes through (2, 4) and (-3, 4). • y = 4

  13. Write the equation of a line given... 1. Slope and y-intercept 2. Graph 3. Slope and one point 4. Two points 5. x - and y-intercepts

  14. m = -3, b = 1 m = -2, b = -4 m = 0, b = 10 m = 1, b = 0 m = 0, b = 0 Find the Equation of the LineGiven the Slope and y-intercept • Substitute m and b intoy = mx + b y = -3x + 1 y = -2x - 4 y = 0x + 10, y = 10 y = 1x + 0, y = x y = 0x + 0, y = 0

  15. Find the Equation of a LineGiven the Graph • Find the y-intercept from the graph. • Count the slope from the graph. • To write the equation of the line, substitute the slope and y-intercept in the slope-intercept form of the equation.

  16. +2 +3 x y Example 1 • b = -3 • m = • y = x - 3

  17. -2 +1 x y Example 2 • b = 1 • m = • y = x + 1

  18. x y Example 3 • b = 4 • m = 0/1= 0 • y = 0x + 4, y = 4

  19. Find the Equation of a LineGiven the Point and the Slope • Use the Point-Slope Formula: • is the given point • Substitute m and into the formula

  20. Example • Write the equation of the line with slope = -2 and passing through the point (3, -5). • Substitute m and into the Point-Slope Formula.

  21. Find the Equation of the LineGiven Two Points • Calculate the slope of the two points. • Use one of the points and the slope to substitute into the Point-Slope formula.

  22. Example • Write the equation of the line that goes through the points (3, 2) and (5, 4).

  23. Find the Equation of the LineGiven the x- and y - intercepts • Write the intercepts as ordered pairs. The x-intercept 4 is the ordered pair (4, 0). The y-intercept -2 is the ordered pair (0, -2). • Calculate the slope. • Substitute the slope and the y-intercept (b) into the slope-intercept formula.

  24. Example Write the equation of the line with x-intercept 3 and y-intercept 2. x-intercept 3 = (3, 0); y-intercept 2 = (0, 2) Slope: y = mx + b

  25. What do you know about the slope of perpendicular lines? Perpendicular lines have slopes that are opposite reciprocals! What is the slope of the line perpendicular to y = -2x + 4? m = 1/2 Perpendicular Lines

  26. What do you know about the slope of parallel lines? Parallel lines have the SAME SLOPE! What is the slope of the line parallel to y = -2x + 4? m = -2 Parallel Lines

  27. Write the equation of a line parallel to 2x – 4y = 8 and containing (-1, 4): – 4y = - 2x + 8 y = 1x - 2 2 Slope = 1 2 y - 4 = 1(x + 1) 2

  28. Write the equation of a line perpendicular to y = -2x + 3and containing (3, 7): Original Slope= -2 Perpendicular Slope = 1 2 y - 7 = 1(x - 3) 2

  29. Write the equation of a line perpendicular to 3x – 4y = 8 and containing (-1, 4): -4y = -3x + 8 y - 4 = -4(x + 1) 3 Slope= 3 4 Perpendicular Slope = -4 3

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