# The equation 3 x 5 y 30 could be a budget constraint

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The equation 3 x + 5 y = 30 could be a budget constraint, for someone spending \$30, facing prices of \$3/unit for x and \$5/unit for y , with x and y representing the number of units of the two goods purchased. In this case, the vertical intercept is the maximum number of units of y that can be purchased (= 30/5 = 6), the horizontal intercept is the maximum number of units of x that can be purchased (= 30/3 = 10), and the slope (= –3/5) indicates that at any point, if 3/5 of a unit of y is given up , 1 additional unit of x can be purchased with the money saved. Regardless of the economic interpretation we give to a particular linear function, however, the above mathematical relationships will hold true. 2. Exercises 1. Convert each of the following equations into the form y = a + b x, where a is the ver- tical intercept and b is the slope, give the horizontal intercept for each, and graph all of them on a single diagram. (a) 3 = ( y – 6)/( x – 6) (b) 3 = ( y + 6)/(6 – x ) (c) 9 x + 3 y = 36 (d) 9 x – 3 y = 36 (e) x = 4 + (1/3) y (f) x = 4 – (1/3) y (g) x = –4 + (1/3) y (h) x = –4 – (1/3) y 2. Graph the following equations on the same diagram. For each equation, give the value of y when x = 10 and x = 20 and the value of x when y = 5 and y = 10. (a) y = 10 – 0.5 x (b) y = 10 – x (c) y = 10 – 2 x (d) y = 20 – x (e) y = 20 – 0.5 x M2-4 MATH MODULE 2: LINEAR EQUATIONS

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3. Give the equations for the following in slope-intercept form, y = a + b x, graph them all on a single diagram and give the horizontal and vertical intercepts where appro- priate: (a) Slope = +3, line passes through (10, 40) (b) Slope = +2, line passes through (5, 30) (c) Slope = +1, line passes through (–30, 0) (d) Slope = 0, line passes through (20, 40) (e) Slope = –1, line passes through (40, 10) (f) Slope = infinity [vertical], line passes through (10, 10). 4. Given the following data for sets of 2 points on some straight lines, calculate the slopes of the lines and give their equations in slope-intercept form, y = a + b x, and graph them in a single diagram: (a) A 1 (20, 10), A 2 (0, 30) (b) B 1 (10, 12.5), B 2 (40, 5) (c) C 1 (10, 5), C 2 (30, 15) (d) D 1 (10, 0), D 2 (15, 5) (e) E 1 (5, 10), E 2 (15, 10) MATH MODULE 2: LINEAR EQUATIONS M2-5
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