Explanation Given Information: The provided graph of shaded triangle is Each triangle has its base on one of the axes and the area of a triangle is one-half the length of its base times its height. Formula Used: Consider the pair of straight lines: L : y = m x + c M : y = n x + b To evaluate the point of intersection of these lines, Step 1. First, convert the equations to slope-intercept form. Step 2. Then, equate the two expressions for y . Step 3. Solve for x and y . To find the area of the shaded region. Consider the formula: Area = 1 2 × base × height . Calculation: Consider the pair of straight lines, { 3 x + 4 y = 24 2 x − 4 y = − 4 Convert the first equation into the slope-intercept form by subtracting 3 x from both the sides and dividing the equation by 4 as follows: y = − 3 4 x + 6 Convert the second equation into the slope-intercept form by subtracting 2 x from both the sides and dividing the equation by − 4 as follows: y = 1 2 x + 1 Compare the two equations for y and compute x as follows: − 3 4 x + 6 = 1 2 x + 1 Add 3 4 x to both sides. 6 = 1 2 x + 3 4 x + 1 Subtract 1 from both sides. 5 4 x = 5 Divide both sides by 4 5 x = 4 Substitute x = 4 in y = 1 2 x + 1 and solve for y , y = 1 2 x + 1 = 1 2 × 4 + 1 = 3 The point of intersection is ( 4 , 3 )

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Chapter 1.3, Problem 30E

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To calculate: The area of the shaded region. The provided figure is shown below:

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Chapter 1.3, Problem 29E

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The area of the shaded region. The provided figure is shown below:

Solution Summary: The author calculates the area of the shaded region by converting the equations to slope-intercept form, comparing the two expressions for x and y.

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