Algebra I and Geometry Concepts Flashcards

A local gym charges a one-time registration fee and a monthly membership fee. The table below shows the total cost (y) for a certain number of months (x). | Months (x) | Total Cost (y) | |------------|----------------| | 2 | $130 | | 4 | $210 | | 6 | $290 | Which of the following linear equations models the total cost for x months of membership?

Answer: y = 40x + 50

To find the linear equation y = mx + b, first find the slope (m), which is the monthly fee. Using two points from the table, (2, 130) and (4, 210), the slope m = (210 - 130) / (4 - 2) = 80 / 2 = $40 per month. Now, substitute the slope and one of the points into the slope-intercept form to find the y-intercept (b), which is the registration fee. Using (2, 130): 130 = 40(2) + b. 130 = 80 + b. b = 50. Therefore, the equation is y = 40x + 50.

A rectangular garden has a length of (x + 7) feet and a width of (x - 2) feet. If the area of the garden is 50 square feet, which equation can be used to find the value of x?

Answer: x² + 5x - 14 = 50

The area of a rectangle is calculated by multiplying its length by its width. In this scenario, the area is (x + 7)(x - 2). To set up the equation, expand the binomials: (x + 7)(x - 2) = x(x) + x(-2) + 7(x) + 7(-2) = x² - 2x + 7x - 14 = x² + 5x - 14. Since the area is given as 50 square feet, the resulting equation is x² + 5x - 14 = 50.

In the figure below, line m is parallel to line n, and they are intersected by transversal line t. If the measure of angle 2 is 115°, what is the measure of angle 7?

Answer: 65°

Angle 2 and angle 6 are corresponding angles, so they are congruent; m∠6 = 115°. Angle 6 and angle 7 are supplementary angles because they form a linear pair. Therefore, their sum is 180°. To find the measure of angle 7, subtract the measure of angle 6 from 180°: 180° - 115° = 65°. Alternatively, Angle 2 and Angle 3 are supplementary (180 - 115 = 65). Angle 3 and Angle 7 are alternate interior angles, so they are congruent. Thus, m∠7 = 65°.

A city park is shaped like a right triangle. The two legs of the triangle measure 8 miles and 15 miles. A new path is being built along the hypotenuse of the triangle. What is the length of the new path?

Answer: 17 miles

This problem requires the use of the Pythagorean theorem, a² + b² = c², where 'a' and 'b' are the legs of the right triangle and 'c' is the hypotenuse. Substitute the given values: 8² + 15² = c². Calculate the squares: 64 + 225 = c². Add the results: 289 = c². Finally, find the square root of 289 to solve for c: c = √289 = 17. The length of the path is 17 miles.

Which of the following are the solutions to the quadratic equation x² - 3x - 10 = 0?

Answer: x = 5 and x = -2

To solve the quadratic equation, we need to factor the trinomial x² - 3x - 10. We are looking for two numbers that multiply to -10 and add up to -3. These numbers are +2 and -5. Therefore, the factored form is (x + 2)(x - 5) = 0. According to the zero-product property, we set each factor equal to zero to find the solutions. So, x + 2 = 0 gives x = -2, and x - 5 = 0 gives x = 5.

A homeowner is calculating the area of a room that has an L-shape. The composite shape can be divided into two rectangles. The first rectangle measures 10 feet by 12 feet. The second, smaller rectangle measures 6 feet by 7 feet. What is the total area of the room?

Answer: 162 square feet

To find the area of a composite shape, you must break it down into simpler shapes, find the area of each part, and then add them together. The area of the first rectangle is length × width = 10 ft × 12 ft = 120 square feet. The area of the second rectangle is 6 ft × 7 ft = 42 square feet. The total area is the sum of the areas of the two rectangles: 120 sq ft + 42 sq ft = 162 square feet.