# Digital SAT Math Practice Questions – Advanced : Linear functions

## SAT MAth Practice questions – all topics

• Algebra Weightage: 35%  Questions: 13-15
• Linear equations in one variable
• Linear equations in two variables
• Linear functions
• Systems of two linear equations in two variables
• Linear inequalities in one or two variables

## SAT MAth and English  – full syllabus practice tests

[Calc]  Question  Hard

Some values of $$x$$ and their corresponding values of $$f(x)$$ for the linear function $$f$$ are shown in the table. What is the value of $$f(6)$$ ?
A) 7
B) 8
C) 9
D) 10

Ans:A

To find the value of $$f(6)$$, we can first identify the slope of the linear function $$f$$. We’ll use the points $$(-2, -5)$$ and $$(4, 4)$$ from the table.

The slope $$m$$ of a linear function is given by:
$m = \frac{{\Delta y}}{{\Delta x}}$

Using the given points:
$m = \frac{{4 – (-5)}}{{4 – (-2)}} = \frac{{4 + 5}}{{4 + 2}} = \frac{9}{6} = \frac{3}{2}$

Now that we have the slope, we can use the point-slope form of a linear equation to find the equation of the line. Let’s use the point $$(4, 4)$$:
$y – y_1 = m(x – x_1)$
$y – 4 = \frac{3}{2}(x – 4)$

Now, let’s find the value of $$f(6)$$ using this equation:
$f(6) = \frac{3}{2}(6 – 4) + 4 = \frac{3}{2}(2) + 4 = 3 + 4 = 7$

So, the value of $$f(6)$$ is $$7$$.

Therefore, the correct answer is:
$\boxed{\text{A) 7}}$

[calc]  Question Hard

Line $$k$$ has a slope of $$-\frac{4}{5}$$ and an $$x$$-intercept of $$\left(\frac{r}{2}, 0\right)$$, where $$r$$ is a constant. What is the $$y$$-coordinate of the $$y$$-intercept of line $$k$$ in terms of $$r$$ ?
A) $$-\frac{2 r}{5}$$
B) $$\frac{2 r}{5}$$
C) $$-\frac{5 r}{8}$$
D) $$\frac{5 r}{8}$$

Ans: B

Line $$k$$ has a slope of $$-\frac{4}{5}$$ and an $$x$$-intercept of $$\left(\frac{r}{2}, 0\right)$$. We need to find the $$y$$-coordinate of the $$y$$-intercept in terms of $$r$$.

The equation of a line in slope-intercept form is:
$y = mx + b$
where $$m$$ is the slope and $$b$$ is the $$y$$-intercept.

Given the slope $$m = -\frac{4}{5}$$ and the $$x$$-intercept $$\left(\frac{r}{2}, 0\right)$$, we can use the point-slope form to find the $$y$$-intercept $$b$$.

Using the point-slope form equation:
$y – y_1 = m(x – x_1)$
where $$(x_1, y_1) = \left(\frac{r}{2}, 0\right)$$, we have:
$y – 0 = -\frac{4}{5}\left(x – \frac{r}{2}\right)$
$y = -\frac{4}{5}x + \frac{4}{5} \cdot \frac{r}{2}$
$y = -\frac{4}{5}x + \frac{2r}{5}$

Therefore, the $$y$$-intercept $$b$$ is:
$b = \frac{2r}{5}$

So, the correct answer is:
$\boxed{\frac{2r}{5}}$

[No calc]  Question Hard

Line k in the xy-plane has slope $$\frac{-2p}{5}$$and y-intercept (0,p), where p is a positive constant. What is the x-coordinate of the x-intercept of line k ?

Ans: 5/2, 2.5

The equation of line $$k$$ is given as $$y = -\frac{2p}{5}x + p$$. To find the $$x$$-intercept, we set $$y = 0$$ and solve for $$x$$:

$0 = -\frac{2p}{5}x + p$

$\frac{2p}{5}x = p$
$x = \frac{5}{2}$

So, the $$x$$ coordinate of the $$x$$-intercept of line $$k$$ is $$\frac{5}{2}$$.

[Calc]  Question  Hard

The table shows the list price, discount, and installation fee for tires from four different car repair stores. Assume there is no sales tax and the information in the table is for tires of the same brand and size

Store W’s total expenses for selling and installing 4 tires is $100. Which function represents the profit p(a), in dollars, from selling and installing 4 tires to which the store’s discount is applied? (profit = total amount of money received – expenses) A) p(a) = 3a + 50 B) p(a) = 3a − 50 C) p(a) = 4a + 50 D) p(a) = 4a − 50 ▶️Answer/Explanation B) p(a) = 3a − 50 For Store W, the list price per tire is $$\a$$. The store offers a “Buy 3 tires at list price and get the 4th tire free” discount. The installation fee for all 4 tires is $$\50$$. Calculate the total amount received from selling the tires: $\text{Total amount received} = 3 \times \a + 0 \times \a = 3a$ Add the installation fee: $\text{Total amount received with installation} = 3a + 50$ The total expenses for selling and installing 4 tires is $$\100$$. Therefore, the profit $$p(a)$$ is given by: $p(a) = (\text{Total amount received with installation}) – \text{Expenses}$ $p(a) = (3a + 50) – 100$ $p(a) = 3a – 50$ [Calc] Question Hard A company spent a total of $$\ 9000$$ on digital and print ads. The ratio of the money spent on digital ads to the money spent on print ads was 1 to 3 . How much money, in dollars, did the company spend on digital ads? (Disregard the \$ sign when entering your answer. For example, if your answer is $$\ 4.97$$, enter 4.97)

2250

Let’s represent the money spent on digital ads as $$d$$ dollars and the money spent on print ads as $$3d$$ dollars, based on the given ratio.

Given that the total amount spent is 9000, we can set up the equation: $d + 3d = 9000$ Solve for $$d$$: $4d = 9000$ $d = \frac{9000}{4}$ $d = 2250$ Therefore, the company spent $$\2250$$ on digital ads. [No calc] Question Hard The complete graph of the function f is shown in the xy-plane. What is the y-intercept of the graph of y = f (x + 2) ? A)(0 , 3) B)(0 , 2) C)(0 , 1) D)(0 , 0) ▶️Answer/Explanation A)(0 , 3) 1. Understanding the transformation: The function $$y=f(x+2)$$ represents a horizontal shift of the graph of $$y=f(x)$$ by 2 units to the left. 2. Y-intercept definition: The $$y$$-intercept is the point where the graph intersects the $$y$$-axis. This occurs when $$x=0$$. 3. Finding the new $$y$$-intercept: For the transformed function $$y=f(x+2)$$, the $$y$$-intercept occurs at $$x=0$$. $y=f(0+2)=f(2)$ [Calc] Question Hard Line l is shown in the xy-plane, and the point with coordinates (2, c) is on line l. What is the value of c ? ▶️Answer/Explanation Ans: 12/5, 2.4 lets find equation of line first which is passing through (0,4) and (5,0). $m=\frac{y_2-y_1}{x_2-x_1}=\frac{0-4}{5-0}=\frac{-4}{5}$ Now, let’s choose one of the points, say $$(0,4)$$, and substitute the coordinates and the slope into the point-slope form: \begin{aligned} & y-4=\frac{-4}{5}(x-0) \\ & y-4=-\frac{4}{5} x \end{aligned} Now, let’s simplify and rewrite the equation in slope-intercept form $$(y=m x+b)$$ : $y=-\frac{4}{5} x+4$ The equation of the line is given as $$y=-\frac{4}{5} x+4$$. Substituting $$x=2$$ and $$y=c$$, we get: \begin{aligned} & c=-\frac{4}{5}(2)+4 \\ & c=-\frac{8}{5}+4 \\ & c=-\frac{8}{5}+\frac{20}{5} \\ & c=\frac{12}{5} \end{aligned} So, the value of $$c$$ is $$\frac{12}{5}$$. Therefore, the point with coordinates $$\left(2, \frac{12}{5}\right)$$ lies on the line. [Calc] Question Hard What is the x-coordinate of the x-intercept of the line with equation $$\frac{5}{4}x + \frac{2}{3}y=1$$ when it is graphed in the xy-plane? ▶️Answer/Explanation 0.8 To find the $$x$$-intercept of the line with equation $$\frac{5}{4}x + \frac{2}{3}y = 1$$, we set $$y = 0$$ and solve for $$x$$. $\frac{5}{4}x + \frac{2}{3}(0) = 1$ $\frac{5}{4}x = 1$ $x = \frac{4}{5} \times 1$ $x = \frac{4}{5}$ So, the $$x$$-coordinate of the $$x$$-intercept is $$\frac{4}{5}$$. [Calc] Question Hard $$|2 x+6|+4=8$$ What is the sum of the solutions to the given equation? A) -6 B) -3 C) 0 D) 8 ▶️Answer/Explanation A Questionsa x-4(3+2 x)=-12$In the equation above,$a$is a constant. For what value of$a$does the equation have infinitely many solutions? A. -8 B. -2 C. 2 D. 8 ▶️Answer/Explanation Ans: D Questions Hongbo sold$x$cell phones in 2013. The number of cell phones he sold in 2014 was$128 \%$greater than in 2013, and the number of cell phones he sold in 2015 was 29\% greater than in 2014. Which of the following expressions represents the number of cell phones Hongbo sold in 2015? A.$(0.29)(1.28 x)$B.$(0.29)(2.28 x)$C.$(1.29)(1.28 x)$D.$(1.29)(2.28 x)$▶️Answer/Explanation Ans: D Question$p=9 n-(2 n+k)$The profit$p$, in dollars, from producing and selling$n$units of a certain product is given by the equation above, where$k$is a constant. If 200 units are produced and sold for a profit of$\$1275$, what is the value of $k$ ?

Ans: 125

Questions

If $|2 x+3|=5$ and $|3 y-3|=6$, what is one possible value of $|x y|$ ?

Ans: 1,3,4,12

Question

If $\frac{4 x+4 x+4 x+4 x}{4}=4$, what is the value of $4 x$ ?
A. 16
B. 4
C. 1
D. $\frac{1}{4}$

Ans: B

Questions

The table above shows two pairs of values for the linear function $$f$$. The function can be written in the form $$f(x)=ax+b$$, where $$a$$ and $$b$$ are constants. What is the value of $$a+b$$?

Ans: 3.25, 13/4

Question

Students in a science lab are working in groups to build both a small and a large electrical circuit. A large circuit uses 4 resistors and 2 capacitors, and a small circuit uses 3 resistors and 1 capacitor. There are 100 resistors and 70 capacitors available, and each group must have enough resistors and capacitors to make one large and one small circuit. What is the maximum number of groups that could work on this lab project?

Ans: 14

Question

$|5-x|=4$

The value of one solution to the equation above is 1. What is the value of the other solution?

Ans: 9

Question

$1.2(h+2)=2 h-1.2$

What value of $h$ is the solution of the equation above?

Ans: $9 / 2,4.5$