Sec. 2-2: Linear Equations 9/19/17

Linear Function (equation): A function whose graph is a line. Dependent Variable: y “Dependent” since the value of y depends on x. Independent Variable: x It depends upon nothing.

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)

Standard Form of a Linear Equation: Ax + By = C 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.

Slope: given two points (x1, y1) & (x2, y2) … m = y2 – y1 = ∆y = “rise” x2 – x1 ∆x “run” Find the slope of the line that goes through the points: (2/3, 1) and (-3, ½) m = 1 – ½ = ½ = ½ 2/3 + 3 2/3 + 9/3 11/3 m = ½ ● 3/11 = 3/22

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 (b) first From that point, move according to the slope: “rise” “run”

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

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) (x1, y1) are the coordinates; m is the slope 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 stuck on the “STANDARD FORM”. When you need to write an equation you can always use the POINT-SLOPE formula. y – y1 = m(x – x1) y – 2 = -3(x + 1) y – 2 = -3x – 3 Then manipulate the equation to put it in STANDARD FORM. 3x + y = -1

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.

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 to determine the slope. Use the Point-Slope Formula to write the equation, plugging in the desired slope & the given point. Put the equation in the desired format.

We need a ┴ slope so new 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 new m = -3/2 2. Use y – y1 = m(x – x1) and plug in the new slope and the given point. y – 9 = -3/2(x + 5) that’s it! {It didn’t specify a certain form.}

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