9-8 Systems Of Linear And Quadratic Equations

1) What is the solution of the system of equations shown below?

y space equals space x squared minus 5 x plus 18 y space equals space 4 x space plus thin space 4

(-2, -4) and (-7 and -24)

(0, 4) and (2, 12)

(2, 12) and (7, 32)

(4, 20) and (5, 25)

The solution of the system of equations is (2, 12) and (7, 32). These points satisfy both equations in the system, meaning that when the x-coordinate is substituted into the first equation and the y-coordinate is substituted into the second equation, both equations are true. Therefore, these points are the solutions to the system of equations.

Explanation

The solution of the system of equations is (2, 12) and (7, 32). These points satisfy both equations in the system, meaning that when the x-coordinate is substituted into the first equation and the y-coordinate is substituted into the second equation, both equations are true. Therefore, these points are the solutions to the system of equations.

2) What is the solution of the system of equations shown below?

y space equals space x space minus space 2 y space equals space x squared minus space 8 x space plus thin space 6

(-1, -3) and (-8, -10)

(0, -2) and (5, 3)

(2, 0) and (-8, -10)

(1, -1) and (8, 6)

The solution of a system of equations is the set of values that satisfy all of the equations in the system. In this case, the given answer (1, -1) and (8, 6) represents two points that satisfy both equations in the system. This means that when the x and y values of these points are plugged into the equations, both equations are true. Therefore, these points are the solution to the system of equations.

Explanation

The solution of a system of equations is the set of values that satisfy all of the equations in the system. In this case, the given answer (1, -1) and (8, 6) represents two points that satisfy both equations in the system. This means that when the x and y values of these points are plugged into the equations, both equations are true. Therefore, these points are the solution to the system of equations.

3) Which point is a solution to the system shown?

y space equals space x squared plus 4 x plus 3 y space equals space 2 x plus 6

(-3, 0)

(1, -8)

(3, 0)

(0, -3)

The point (-3, 0) is a solution to the system shown because it satisfies both equations in the system.

Explanation

The point (-3, 0) is a solution to the system shown because it satisfies both equations in the system.

4) Which coordinate is a solution to the following system?

y space equals space x squared plus 2 x minus 2 y space equals space x plus 10

(4, 14)

(3, 13

(2, 12)

(-3, 7)

The given system of coordinates consists of four points. The answer states that (3, 13) is a solution to this system. This means that when we plug in the values of x=3 and y=13 into the equations of the system, they satisfy all the equations.

Explanation

The given system of coordinates consists of four points. The answer states that (3, 13) is a solution to this system. This means that when we plug in the values of x=3 and y=13 into the equations of the system, they satisfy all the equations.

5) An architect makes a drawing of a parabolic-shaped arch with a linear support intersecting it in two places. The parabola can be modeled by the function

y space equals space x squared minus space 5 x space plus space 10

. THe line intersects the parabola when x = 2 and x = 4. What is the equation of the line?

Y = x- 6

Y = x - 2

Y = x + 2

Y = x + 6

The line intersects the parabola at x = 2 and x = 4. To find the equation of the line, we need to determine the y-intercept. We can substitute the x-coordinate of one of the intersection points into the equation of the parabola to find the corresponding y-coordinate. When x = 2, the equation of the parabola gives y = 2 + 2 = 4. Therefore, the y-intercept is 4. Since the line intersects the y-axis at (0, 2), the slope of the line is 1. Using the slope-intercept form of a line, y = mx + b, where m is the slope and b is the y-intercept, we have y = x + 2. Hence, the equation of the line is y = x + 2.

Explanation

The line intersects the parabola at x = 2 and x = 4. To find the equation of the line, we need to determine the y-intercept. We can substitute the x-coordinate of one of the intersection points into the equation of the parabola to find the corresponding y-coordinate. When x = 2, the equation of the parabola gives y = 2 + 2 = 4. Therefore, the y-intercept is 4. Since the line intersects the y-axis at (0, 2), the slope of the line is 1. Using the slope-intercept form of a line, y = mx + b, where m is the slope and b is the y-intercept, we have y = x + 2. Hence, the equation of the line is y = x + 2.