Sec. 2-2: Linear Equations

Linear Function (equation): A function whose graph is a line. Dependent Variable: (y) Since the value of y depends on(or is determined by) “x”. Independent Variable: x “X” controls what happens.

Y-Intercept: (0, #) A point where a line crosses the y-axis. Found by putting zero in for the x value. X-Intercept: (#, 0) A point where a line crosses the x-axis. Found by putting zero in for the y value. Given y = 5x + 7 find the intercepts. The x-Int. is (-7/5, 0) the y-Int. is (0, 7) Video Help

Standard Form of a Linear Equation: Ax + By = C No fractions “A” must be POSITIVE x & y are on the SAME side of the equation Finding x- & y-intercepts is easiest when the equation is in this format. Video Help

Slope: m = y1 – y2 = ∆y = “rise” x1 – x2 ∆x “run” Find the slope of the points: (2, 1) and (-3, ½) m = 1 – ½ = ½ = 1/2 2 + 3 2 + 3 5 m = = = Video Help End of day 1

Slope-Intercept Form: y = mx + b Where m is the slope and b is the y-intercept Easy format to graph with: Graph the y-intercept first From that point, move according to the slope: “rise” “run” Video Help

To write an equation with the slope-intercept form, you would need the SLOPE & the y-intercept. i.e. Write the equation that contains the points (-2, 0) & (0, 5). *** You would first determine the slope & use the y-intercept (0, 5) m = 5 – 0 = 5 so y = 5/2x + 5 0 + 2 2 Video Help

9. Point-Slope Form: Used to write an equation of a line if you’re given a “point” on the line, and the “slope” of the line. y – y1 = m (x – x1) Plug your information into the highlighted variables and solve the equation for the desired format.

Write the standard form equation of a line with slope -3 that goes through the point (-1, 2). *** Don’t get hung up on the “STANDARD FORM”. . . When you need to write an equation you will always use the POINT-SLOPE formula. y – y = m(x – x) y – 2 = -3(x + 1) y – 2 = -3x – 3 Then manipulate the equation to put it in STANDARD FORM. 3x + y = -1 End of day 2 Video Help

Lines that never intersect. Slopes are the same. Parallel Lines: // Lines that never intersect. Slopes are the same. Perpendicular Lines: ┴ Lines that intersect at 90° angles. Slopes are OPPOSITE RECIPROCALS. Identify the following lines as // or ┴ or neither. Video Help

y = 3x – 2 y = 3x -12 // because both slopes are 3 2y = -x + 5 y = 2x + 4 ┴ because the 1st slope is -1/2 & the 2nd is 2 3x – 2y = -8 x + y = 1 Neither because the 1st slope is 3/2 & the 2nd is -1

Writing equations of // & ┴ lines Determine the desired slope. You may have to manipulate the original equation around to pick off the slope. Use the Point-Slope Form to write the equation, plugging in the desired slope & the given point. Put the equation in the desired format.

We need a ┴ slope so m ┴ = -3/2 Write an equation of a line ┴ 2x – 3y =7 and that goes through the point (-5, 9). 1. First, determine the current line’s slope: (solve for y) y = 2/3x - 7/3 so m = 2/3 We need a ┴ slope so m ┴ = -3/2 2. Use y – y = m(x – x) and plug in the new slope and the given point.