9th Grade Math Final

Consider the system of equations

3x + 4y = 0

-3x +-4y = 2

Adding the two equations side-by-side and simplifying yields 0 = 2.

Which of the following can be concluded about the system of equations?

• It has a unique solution (2, 0).

• It has exactly two solutions (2, 0) and (0, 2).

• It has infinitely many solutions.

• It has no solution.

Answer explanation

Since both sides of the equation do not equal each other, this is how you can tell that a problem has no solution.

Tags

CCSS.8.EE.C.8B

On a number line, the coordinates of A and B are -3 and b, respectively. If the coordinate of the midpoint M of AB is 5, what is the value of b?

• 12

• 15

• 13

• 4

Answer explanation

If you subtract a point from the midpoint you are able to determine the length of that segment. Midpoint(M) - point of segment (A) = AM

Then use the property of midpoints:

AM=1/2(AxB)

[M is the midpoint of AB]

5-(-3) = 1/2(b-(-3)) [Substitution]

8= 1/2 (b+3)

8= 1/2b + 3/2

multiply both sides of the equation by 2

8x2=2(1/2b) + 2(3/2)

16=b + 3

16-3 = b

13 = b

Tags

CCSS.HSG.GPE.B.6

Refer to the graph above. The shaded region is the solution set of:

• $y\le2x+2$

• $y\ge2x+2$

• $y<2x+2$

• $y>2x+2$

Answer explanation

Locate 2 points on the graph of the line. For example: (-1, 0) & (0,2).

Use these points to figure out the slope of the line. $\frac{y_2-y_1}{x_2-x_1}=\frac{2-0}{0-\left(-1\right)}=\frac{2}{1}$
Then, find the point that is your y-intercept, [meaning where the line intersects the y-axis] (0,2) Now we know the equation is y=2x+2.
Since the line is a solid line and not dotted, we know that it is either $\le or\ge$ .
Finally, we try one of the ordered pairs we picked with both types of inequalities.

The only one that works and matches with the shaded region is $y\le2x+2$

Tags

CCSS.HSA.REI.D.12

A boat travels with the current at 15km/h and travels against the current at 7km/h. What is the speed of the boat in still water, and what is the rate of the current?

• Speed: 15km/h

  Current: 7km/h

• Speed: 13 km/h

  Current: 6 km/h

• Speed: 11 km/h

  Current: 4 km/h

• Speed: 10 km/h

  Current: 5 km/h

Answer explanation

You will need two equations. One that shows the boat's speed with the current, and one that shows the boat speed against the current.

b+c=15(this equation shows the speed of the boat with the current equals 15km/h)

b-c=15(this equation shows the speed of the boat against the current equal 7km/h)

b + c =15

b - c = 7

Use the elimination method to solve for b.

This yields 2b = 22

b = 22/2

b = 11

The boat's speed in still water is 11km/hr.

Plugin 11 km/hr into the first equation and solve for c.

11 + c = 15

c = 15 - 11

c = 4

The rate of the current is 4 km/hr

Tags

CCSS.8.EE.C.8C

Triangle KLP is _________________________ triangle

• right

• obtuse

• acute

Tags

CCSS.4.G.A.2

Triangle ALK is _________________________ triangle.

• right

• obtuse

• acute

Tags

CCSS.4.G.A.2

Triangle APK is _________________________ triangle

• Right

• Obtuse

• Acute

Tags

CCSS.4.G.A.2

Solve for x.

• 34

• 56

• 42

• 24

Answer explanation

The sum of he measures of the interior angles of a five sided figure is 540 degrees.

4x-13+70+90+2x+10+3x+5=540

Combine like terms

9x+162=540

9x=540-162

9x=378

x=378/9

x=42

Tags

CCSS.HSA.CED.A.1

CCSS.HSA.REI.A.1

CCSS.HSA.REI.B.3

Given triangle ABC with vertices A(-2,-2), B(1,2), and C(4,-2) Find the Perimeter.

• 6

• 12

• 16

• 20

Answer explanation

$AB=\sqrt{\left(1-2\right)^2\ +\ \left(2-2\right)^2}$ =5
$BC=\sqrt{\left(4-1\right)^2+\left(-2-2\right)^2}$ =5
$AC=\sqrt{\left(4+2\right)^2+\left(-2+2\right)^2}$ =6
So, 5+5+6=16
The perimeter of the triangle is 16.

Tags

CCSS.HSG.GPE.B.7

Given quadrilateral PQRS with vertices P(1, 2), Q(3, 0), R(0, -3), and S(-2, -1). Which of the following best describes the type of the quadrilateral PQRS?

• rectangle

• trapezoid

• square

• parallelogram

Answer explanation

$\frac{-1-2}{-2-1}=1$ Slope of PS

$\frac{0-2}{3-1}=\frac{-2}{2}=-1$ Slope PQ

$\frac{-3-0}{0-3}=1$ Slope QR
$\frac{-1-3}{-2-0}=\frac{2}{-2}=-1$ Slope RS
The slope of adjacent sides of PQRS are negative reciprocals of each other. That means that any two adjacent sides are perpendicular. This means PQRS is a rectangle.

Tags

CCSS.6.G.A.3