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Transcript
Linear Equations in Two Variables MATH 109 - Precalculus S. Rook Overview • Section 1.3 in the textbook: – Graphing a linear equation by using its slope – Finding the slope of a line – Writing Linear Equations – Parallel and Perpendicular Lines 2 Graphing a Linear Equation by Using its Slope Graphing Lines in Slope-Intercept Form • Slope-Intercept Form of a line: y = mx + b where m is the slope and the y-intercept is (0, b) • To graph a line using its slope and y-intercept: – Solve the equation for y if necessary – Plot the y-intercept (0, b) – Use rise over run with the slope to get 2 or 3 more points 4 Other Lines • Two other type of lines: – When a linear equation is only in 1 variable (x or y) – Vertical lines have the form x = a (a is a constant): • m = Ø (undefined) • To sketch, find x = a on the x-axis and draw a vertical line – Horizontal lines have the form y = b (b is a constant) • m=0 • To sketch, find y = b on the y-axis and draw a horizontal line 5 Graphing a Linear Equation by Using its Slope (Example) Ex 1: Sketch the graph of the linear equation using its slope and y-intercept: a) x y 5 2 b) y x 3 3 6 Graphing Other Lines (Example) Ex 2: Sketch the line: a) y + 3 = 2 b) x = 4 7 Finding the Slope of a Line Definition and Properties of Slope • Slope (m): the ratio of the change in y (Δ y) and the change in x (Δ x) – Quantifies (puts a numerical value on) the “steepness” of a line • Given 2 points on a line, we can find its slope: y y2 y1 m x x2 x1 9 Sign of the Slope of a Line • To determine the sign of the slope, examine the graph of the line from left to right: – Positive if the line rises – Negative if the line drops 10 Finding the Slope of a Line (Example) Ex 3: Find the slope of the line: a) Through (-2, 5) and (4, -5) b) Vertical line through (3, 11) c) Through (0, 8) and (1, 10) 11 Writing Linear Equations Point Slope Formula & Standard Form of a Line • Given the slope and a point on a line, we can construct its equation • Point-slope formula: y – y1 = m(x – x1) – – – – (x1, y1) is any point x and y are variables Very similar to the slope formula Could also use y = mx + b • Standard form: a linear equation in the form Ax + By = C where A, B, and C are constants – Variables to the left and constants on the right – NO fractions or decimals 13 Writing an Equation in Standard Form Ex 4: Write the linear equation of the line in standard form: a) Through (-4, -8) with a slope of -¼ b) Through (2, 0) with a slope of 1 14 Equations of Lines when Given Two Points • We know how to find the slope of a line given two points • Proceed as before – Pick one of the two points to use in the pointslope formula 15 Writing the Equation of a Line Given Two Points (Example) Ex 5: Write the equation of the line in slopeintercept form: a) Through (4, 7) and (8, 9) b) Through (2, 1) and (-3, 6) 16 Equations of Vertical and Horizontal Lines • Recall the slopes of vertical and horizontal lines: – Slope of a vertical line is undefined – Slope of a horizontal line is zero • Use the slope along with the given point to construct the equation of the line: – If the line is vertical, use the x-coordinate of the given point – If the line is horizontal, use the y-coordinate of the given point 17 Equations of Vertical and Horizontal Lines (Example) Ex 6: Write the equation of the line: a) Through (-2, 5) with undefined slope b) Through (4, 13) with a slope of zero 18 Parallel and Perpendicular Lines Slopes of Parallel and Perpendicular Lines • Parallel lines: two lines that have the SAME slope • Perpendicular lines: two lines that have OPPOSITE RECIPROCAL slopes – i.e. the product of the slopes is -1 20 Classifying Lines as Parallel, Perpendicular, or Neither (Example) Ex 7: Determine whether the two lines are parallel, perpendicular, or neither: a) 3x + y = -4 and 6y – 2x = 12 b) y = 2x + 1 and y = -2x – 3 c) 3y – 15x = -6 and y = 5x + 2 21 Equations of Parallel and Perpendicular Lines • Given the equation of a line, we want to find the equation of a second line that is parallel or perpendicular to the first • Slope is not explicitly given – Put the equation of the given line in slope-intercept form • Determine the appropriate slope based on whether the second line is to be parallel or perpendicular to the first • Use the slope and the given point in the point-slope formula or y = mx + b • A vertical line is parallel to another vertical line and a horizontal line is perpendicular to a vertical line – Vice versa for a horizontal line 22 Equations of Parallel and Perpendicular Lines (Example) Ex 8: Write the equation of the line in standard form if possible: a) Perpendicular to y = -3 and passes through (1, -5) b) Parallel to y = ½x + 5 and passes through (4, 2) c) Perpendicular to 3x + y = -1 and passes through (-1, -1) 23 Summary • After studying these slides, you should be able to: – – – – – Graph a line using its slope and y-intercept Graph vertical or horizontal lines Find the slope of a line Write linear equations Classify two lines as being parallel, perpendicular, or neither – Write equations involving parallel and perpendicular lines • Additional Practice – See the list of suggested problems for 1.3 • Next lesson – Functions (Section 1.4) 24