PSAT: Algebra - Linear Equations with two variables

Last week, an interior designer earned a total of $1,258 from consulting for $x$ hours and drawing up plans for $y$ hours. The equation $68x + 85y = 1{,}258$ represents this situation. Which of the following is the best interpretation of 68 in this context?

Line $k$ is defined by $y = 7x + \frac{1}{8}$. Line $j$ is perpendicular to line $k$ in the $xy$-plane. What is the slope of line $j$?

A machine makes large boxes or small boxes, one at a time, for a total of $700$ minutes each day. It takes the machine $10$ minutes to make a large box or $5$ minutes to make a small box. Which equation represents the possible number of large boxes, $x$, and small boxes, $y$, the machine can make each day?

A producer is creating a video with a length of 70 minutes. The video will consist of segments that are 1 minute long and segments that are 3 minutes long. Which equation represents this situation, where $x$ represents the number of 1-minute segments and $y$ represents the number of 3-minute segments?

An employee at a restaurant prepares sandwiches and salads. It takes the employee 1.5 minutes to prepare a sandwich and 1.9 minutes to prepare a salad. The employee spends a total of 46.1 minutes preparing $x$ sandwiches and $y$ salads. Which equation represents this situation?

A line in the $xy$-plane has a slope of $\dfrac{1}{9}$ and passes through the point $(0,14)$. Which equation represents this line?

A certain township consists of a 5-hectare industrial park and a 24-hectare neighborhood. The total number of trees in the township is 4,529. The equation $5x + 24y = 4{,}529$ represents this situation. Which of the following is the best interpretation of $x$ in this context?

Line $p$ is defined by $2y + 18x = 9$. Line $r$ is perpendicular to line $p$ in the $xy$-plane. What is the slope of line $r$?

Question: What is the slope of the graph of $y = \frac{1}{3}(29x + 10) + 5x$ in the $xy$-plane?

The graph of the equation $ax + ky = 6$ is a line in the xy-plane, where $a$ and $k$ are constants. If the line contains the points $(-2, -6)$ and $(0, -3)$, what is the value of $k$?

The equation $40x + 20y = 160$ represents the number of sweaters, $x$, and number of shirts, $y$, that Yesenia purchased for $160. If Yesenia purchased 2 sweaters, how many shirts did she purchase?

The table shows two values of $x$ and their corresponding values of $y$. In the $xy$-plane, the graph of the linear equation representing this relationship passes through the point $\left(\frac{1}{7}, a\right)$. What is the value of $a$?

The $y$-intercept of the graph of $12x + 2y = 18$ in the $xy$-plane is $(0, y)$. What is the value of $y$?

Tony spends $80 per month on public transportation. A 10-ride pass costs $12.50, and a single-ride pass costs $1.50. If $g$ represents the number of 10-ride passes Tony buys in a month and $t$ represents the number of single-ride passes Tony buys in a month, which of the following equations best represents the relationship between $g$ and $t$?

In the xy-plane, a line has a slope of 6 and passes through the point $(0,8)$. Which of the following is an equation of this line?

If the graph of $27x + 33y = 297$ is shifted down 5 units in the xy-plane, what is the y-intercept of the resulting graph?

$5G + 45R = 380$

At a school fair, students can win colored tokens that are worth a different number of points depending on the color. One student won $G$ green tokens and $R$ red tokens worth a total of $380$ points. The given equation represents this situation. How many more points is a red token worth than a green token?

In 2010, a swim club had a total of 35 swimmers, each classified as either advanced or intermediate. From 2010 to 2020, the number of advanced swimmers in the club increased by approximately 53%, and the number of intermediate swimmers in the club increased by approximately 44%. The total number of swimmers in the club increased by approximately 49%. Which equation best represents this situation, where $a$ represents the number of advanced swimmers in the club in 2010 and $b$ represents the number of intermediate swimmers in the club in 2010?

For a camping trip a group bought $x$ one-liter bottles of water and $y$ three-liter bottles of water, for a total of 240 liters of water. Which equation represents this situation?

Naomi bought both rabbit snails and nerite snails for a total of $\$52$. Each rabbit snail costs $\$8$ and each nerite snail costs $\$6$. If Naomi bought 2 nerite snails, how many rabbit snails did she buy?

$2x + y = 37$

In triangle $QRS$, sides $QR$ and $RS$ each have a length of $x$ centimeters and side $SQ$ has a length of $y$ centimeters. The given equation represents this situation. Which of the following is the best interpretation of $37$ in this context?

In the xy-plane, line $\ell$ passes through the point $(0,0)$ and is parallel to the line represented by the equation $y = 8x + 2$. If line $\ell$ also passes through the point $(3,d)$, what is the value of $d$?

$24.5x + 24.75y = 641$

Isabel ordered topsoil and crushed stone, which cost a total of $641, for her garden. The given equation represents the relationship between the number of cubic yards of topsoil, $x$, and the number of tons of crushed stone, $y$, Isabel ordered. How much more, in dollars, did a ton of crushed stone cost Isabel than a cubic yard of topsoil?

Question: What is the slope of the graph of $y = \frac{5x}{13} - 23$ in the $xy$-plane?

What is the y-intercept of the graph of $y = 34x + 81$ in the $xy$-plane?

Line $\ell$ in the $xy$-plane has a slope of $- \dfrac{1}{3}$ and passes through the point $(9,10)$. Which equation defines line $\ell$?

A store sells two different-sized containers of blueberries. The store’s sales of these blueberries totaled 896.86 dollars last month. The equation $4.51x + 6.07y = 896.86$ represents this situation, where $x$ is the number of smaller containers sold and $y$ is the number of larger containers sold. According to the equation, what is the price, in dollars, of each smaller container?

A chemist studying the impact of salt on a process mixes $x$ kilograms of a low-salt mixture, which is 2% salt by weight, with $y$ kilograms of a high-salt mixture, which is 96% salt by weight, to create 24 kilograms of a mixture that is 4% salt by weight. Which equation represents this situation?

Jay walks at a speed of 3 miles per hour and runs at a speed of 5 miles per hour. He walks for $w$ hours and runs for $r$ hours for a combined total of 14 miles. Which equation represents this situation?

The y-intercept of the graph of $y = -6x - 32$ in the $xy$-plane is $(0, y)$. What is the value of $y$?

A shipment consists of 5-pound boxes and 10-pound boxes with a total weight of 220 pounds. There are 13 10-pound boxes in the shipment. How many 5-pound boxes are in the shipment?

A certain apprentice has enrolled in 85 hours of training courses. The equation $10x + 15y = 85$ represents this situation, where $x$ is the number of on-site training courses and $y$ is the number of online training courses this apprentice has enrolled in. How many more hours does each online training course take than each on-site training course?

$y = -4x + 40$

Which table gives three values of $x$ and their corresponding values of $y$ for the given equation?

Davio bought some potatoes and celery. The potatoes cost $0.69 per pound, and the celery cost $0.99 per pound. If Davio spent $5.34 in total and bought twice as many pounds of celery as pounds of potatoes, how many pounds of celery did Davio buy?

An artist paints and sells square tiles. The selling price $P$, in dollars, of a painted tile is a linear function of the side length of the tile $s$, in inches, as shown in the table below.

Which of the following could define the relationship between $s$ and $P$?

Line $p$ is defined by $4y + 8x = 6$. Line $r$ is perpendicular to line $p$ in the $xy$-plane. What is the slope of line $r$?

What is the equation of the line shown in the $xy$-plane above?

What is the y-coordinate of the y-intercept of the graph of $\dfrac{3x}{7} = -\dfrac{5y}{9} + 21$ in the $xy$-plane?

$x + y = 350$

The given equation relates the total number of maple trees, $x$, and the total number of birch trees, $y$, planted in a 14-acre forest preserve. If 245 maple trees were planted in the forest preserve, how many birch trees were planted in the forest preserve?

The equation $y = 0.1x$ models the relationship between the number of different pieces of music a certain pianist practices, $y$, during an $x$-minute practice session. How many pieces did the pianist practice if the session lasted 30 minutes?

In the $xy$-plane, line $t$ passes through the points $(0, 9)$ and $(1, 17)$. Which equation defines line $t$?

Line $k$ is defined by $y = \tfrac{1}{4}x + 1$. Line $j$ is parallel to line $k$ in the $xy$-plane. What is the slope of $j$?

Which of the following is an equation of the graph shown in the $xy$-plane above?

What is the slope of the graph of $10x - 5y = -12$ in the $xy$-plane?

Vivian bought party hats and cupcakes for $\$71$. Each package of party hats cost $\$3$, and each cupcake cost $\$1$. If Vivian bought 10 packages of party hats, how many cupcakes did she buy?

$F = 2.50x + 7.00y$

In the equation above, $F$ represents the total amount of money, in dollars, a food truck charges for $x$ drinks and $y$ salads. The price, in dollars, of each drink is the same, and the price, in dollars, of each salad is the same. Which of the following is the best interpretation for the number 7.00 in this context?

A gardener buys two kinds of fertilizer. Fertilizer A contains 60% filler materials by weight and Fertilizer B contains 40% filler materials by weight. Together, the fertilizers bought by the gardener contain a total of 240 pounds of filler materials. Which equation models this relationship, where $x$ is the number of pounds of Fertilizer A and $y$ is the number of pounds of Fertilizer B?

The graph of $7x + 2y = -31$ in the $xy$-plane has an $x$-intercept at $(a, 0)$ and a $y$-intercept at $(0, b)$, where $a$ and $b$ are constants. What is the value of $\dfrac{b}{a}$?

Figure A and figure B are both regular polygons. The sum of the perimeter of figure A and the perimeter of figure B is 63 inches. The equation $3x + 6y = 63$ represents this situation, where $x$ is the number of sides of figure A and $y$ is the number of sides of figure B. Which statement is the best interpretation of 6 in this context?

The line with the equation $\frac{4}{5}x + \frac{1}{3}y = 1$ is graphed in the xy-plane. What is the x-coordinate of the x-intercept of the line?

Line $r$ is defined by the equation $4x - 9y = 3$. Line $s$ is parallel to line $r$ in the $xy$-plane. What is the slope of line $s$?

How many liters of a 25% saline solution must be added to 3 liters of a 10% saline solution to obtain a 15% saline solution?

Line $k$ is defined by $y = 3x + 15$. Line $j$ is perpendicular to line $k$ in the $xy$-plane. What is the slope of line $j$?

The graph of $9x - 10y = 19$ is translated down $4$ units in the $xy$-plane. What is the x-coordinate of the x-intercept of the resulting graph?

What is the slope of the graph of $y = \frac{1}{4}(27x + 15) + 7x$ in the $xy$-plane?

A total of 2 squares each have side length $r$. A total of 6 equilateral triangles each have side length $t$. None of these squares and triangles shares a side. The sum of the perimeters of all these squares and triangles is 210. Which equation represents this situation?

$3a + 4b = 25$

A shipping company charged a customer $25 to ship some small boxes and some large boxes. The equation above represents the relationship between $a$, the number of small boxes, and $b$, the number of large boxes, the customer had shipped. If the customer had 3 small boxes shipped, how many large boxes were shipped?

A line in the $xy$-plane has a slope of $9$ and passes through the point $(0,-5)$. The equation $y = px + r$ defines the line, where $p$ and $r$ are constants. What is the value of $p$?

At a state fair, attendees can win tokens that are worth a different number of points depending on the shape. One attendee won $S$ square tokens and $C$ circle tokens worth a total of 1,120 points. The equation $80S + 90C = 1{,}120$ represents this situation. How many more points is a circle token worth than a square token?

The equation is $y = x + 4$.

Which table gives three values of $x$ and their corresponding values of $y$ for the given equation?

A.

B.

C.

D.

$4x + 3y = 24$

Mario purchased 4 binders that cost $x$ dollars each and 3 notebooks that cost $y$ dollars each. If the given equation represents this situation, which of the following is the best interpretation of 24 in this context?

The equation $46 = 2a + 2b$ gives the relationship between the side lengths $a$ and $b$ of a certain parallelogram. If $a = 9$, what is the value of $b$?

The table above shows some pairs of x values and y values. Which of the following equations could represent the relationship between x and y?

Line $r$ in the $xy$-plane has a slope of $4$ and passes through the point $(0,6)$. Which equation defines line $r$?

A neighborhood consists of a 2-hectare park and a 35-hectare residential area. The total number of trees in the neighborhood is 3,934. The equation $2x + 35y = 3{,}934$ represents this situation. Which of the following is the best interpretation of $x$ in this context?

A chemist combines water and acetic acid to make a mixture with a volume of 56 milliliters (mL). The volume of acetic acid in the mixture is 10 mL. What is the volume of water, in mL, in the mixture? (Assume that the volume of the mixture is the sum of the volumes of water and acetic acid before they were mixed.)

$x + y = 75$

The equation above relates the number of minutes, $x$, Maria spends running each day and the number of minutes, $y$, she spends biking each day. In the equation, what does the number 75 represent?

$y = 70x + 8$

Which table gives three values of $x$ and their corresponding values of $y$ for the given equation?

A.

B.

C.

D.

The equation $x + y = 1{,}440$ represents the number of minutes of daylight (between sunrise and sunset), $x$, and the number of minutes of non-daylight, $y$, on a particular day in Oak Park, Illinois. If this day has 670 minutes of daylight, how many minutes of non-daylight does it have?

A mixture consisting of only vitamin D and calcium has a total mass of 150 grams. The mass of vitamin D in the mixture is 50 grams. What is the mass, in grams, of calcium in the mixture?

Line $k$ is defined by $y = \frac{17}{7}x + 4$. Line $j$ is parallel to line $k$ in the $xy$-plane. What is the slope of line $j$?

In an article about exercise, it is estimated that a 160-pound adult uses 200 calories for every 30 minutes of hiking and 150 calories for every 30 minutes of bicycling. An adult who weighs 160 pounds has completed 1 hour of bicycling. Based on the article, how many hours should the adult hike to use a total of 1,900 calories from bicycling and hiking?

A store sells two different-sized containers of a certain Greek yogurt. The store’s sales of this Greek yogurt totaled 1,277.94 dollars last month. The equation $5.48x + 7.30y = 1{,}277.94$ represents this situation, where $x$ is the number of smaller containers sold and $y$ is the number of larger containers sold. According to the equation, which of the following represents the price, in dollars, of each smaller container?

ID: 35584081

A batch of banana milkshakes consists of 4 cups of ice cream and 2 bananas and has 1,114 milligrams (mg) of calcium. There is 276 mg of calcium in 1 cup of the ice cream used to make this batch of milkshakes. How much calcium, in mg, is in 1 banana?

A city’s total expense budget for one year was $x$ million dollars. The city budgeted $y$ million dollars for departmental expenses and 201 million dollars for all other expenses. Which of the following represents the relationship between $x$ and $y$ in this context?