Question

20
this is a preview/pencil copy of an online technology enhanced item.
three systems of equations are shown in the table below.
for each system, drag the choice that correctly describes the number of solutions of each system into the appropriate column in the table below.
2x + 2y = 16
4x + 2y = 27
2x + 2y = 8
4y + 4y = 14
2y - 5y = 12
2x + 3y = 13
one solution
no solution
infinitely many solutions
21
which equation represents the line that is perpendicular to the graph of 4x + 3y = 9 and passes through (-2, 3)?
a 3x - 4y = -18
b 3x + 4y = 18
c 3x - 4y = -5
d 3x + 4y = 6
22
a club began with 3 members. each month, each member brought one new member. which function can be used to determine the number of members x months after the club began?
a f(x) = 2x + 3
b f(x) = 3x + 1
c f(x) = 1.5(2)^x
d f(x) = 3(2)^x

Explanation:

Question 21

Step1: Find the slope of the given line

The given line is \(4x + 3y=9\). Rewrite it in slope - intercept form \(y = mx + b\) (where \(m\) is the slope and \(b\) is the y - intercept).
\(3y=-4x + 9\)
\(y=-\frac{4}{3}x + 3\). So the slope of the given line, \(m_1=-\frac{4}{3}\).

Step2: Find the slope of the perpendicular line

If two lines are perpendicular, the product of their slopes is \(- 1\). Let the slope of the perpendicular line be \(m_2\). Then \(m_1\times m_2=-1\).
\(-\frac{4}{3}\times m_2=-1\)
\(m_2=\frac{3}{4}\).

Step3: Use the point - slope form to find the equation of the line

The point - slope form of a line is \(y - y_1=m(x - x_1)\), where \((x_1,y_1)=(-2,3)\) and \(m = \frac{3}{4}\).
\(y - 3=\frac{3}{4}(x + 2)\)
Multiply both sides by 4 to get rid of the fraction: \(4(y - 3)=3(x + 2)\)
\(4y-12 = 3x + 6\)
Rearrange the terms: \(3x-4y=-18\) (or \(3x - 4y=-18\) can be rewritten as \(3x-4y=-18\), which is option A)

Question 22

Step1: Identify the type of function

The problem is about the growth of the number of club members. Initially, there are 3 members. Each month, each member brings one new member. This is an exponential growth situation.

Step2: Analyze the exponential growth formula

The general form of an exponential function is \(f(x)=a(b)^x\), where \(a\) is the initial amount and \(b\) is the growth factor.

• The initial number of members \(a = 3\).
• In the first month, the number of members: each of the 3 members brings 1 new member, so the number of new members is 3, and the total number of members is \(3 + 3=3\times2\).
• In the second month, each of the \(3\times2\) members brings 1 new member, so the number of new members is \(3\times2\), and the total number of members is \(3\times2+3\times2=3\times2\times2 = 3\times(2)^2\).
• So the growth factor \(b = 2\) and the initial amount \(a = 3\). The function is \(f(x)=3(2)^x\), which is option D.

Answer:

s:

1. A. \(3x - 4y=-18\)
2. D. \(f(x)=3(2)^x\)