25 hard SAT Linear equations · two variables questions

$px + 6y = 18$ $2x + qy = 6$ In the given pair of equations, p and q are constants. The graph of this pair of equations in the xy-plane is a pair of parallel lines. If $q = 3$, what is the value of p?

• A$3$
• B$4$
• C$6$
• D$-4$

A rental car company charges customers based on the number of miles driven. The total charge T, in dollars, for renting a car and driving it for m miles is given by $T = 45 + 0.32m$, where 45 is the base rental fee and 0.32 is the per-mile charge. A customer has a budget of $120 for the rental. The maximum number of miles x the customer can drive is given by the equation $0.32x = 75$. Which statement is the best interpretation of 0.32 in this context?

• AThe charge per mile driven is $0.32.
• BThe total budget remaining after the base fee is $0.32.
• CThe charge per mile driven is $0.23.
• DThe base rental fee per mile is $0.32.

In the xy-plane, the graph of the equation $3x - 4y = 24$ has an x-intercept at (p, 0) and a y-intercept at (0, q). What is the value of $p + q$?

• A$-2$
• B$2$
• C$14$
• D$-1$

In the xy-plane, line p has the equation $4x + 7y = 28$. Line q is parallel to line p and passes through the point (3, -4). What is the y-intercept of line q?

• A$\displaystyle -\frac{16}{7}$
• B$-4$
• C$\displaystyle \frac{16}{7}$
• D$4$

In the xy-plane, line r is defined by the equation $5x + 3y = 30$. Line s passes through the origin and is perpendicular to line r. What is the x-coordinate of the point where line s intersects the line $y = 10$?

• A$-6$
• B$6$
• C$10$
• D$\displaystyle \frac{50}{3}$

$3x + ky = 18$ $2x - 5y = 7$ In the given system of equations, k is a constant. The system has no solution. What is the value of k?

• A$\displaystyle -\frac{15}{2}$
• B$\displaystyle \frac{15}{2}$
• C$\displaystyle -\frac{5}{2}$
• D$\displaystyle -\frac{6}{5}$

In the xy-plane, line m is defined by the equation $5x - 2y = 20$. Line n passes through the point (12, -3) and is perpendicular to line m. What is the y-coordinate of the y-intercept of line n?

• A$\displaystyle \frac{9}{5}$
• B$\displaystyle -\frac{9}{5}$
• C$\displaystyle \frac{2}{5}$
• D$9$

A water tank is being filled at a constant rate. The volume V, in gallons, of water in the tank after t minutes is given by $V = 250 + 18t$, where 250 is the initial volume and 18 is the fill rate in gallons per minute. Due to a simultaneous drain, the net volume N, in gallons, after t minutes is given by $N = 250 + 11t$. Which statement is the best interpretation of 11 in this context?

• AThe net rate of volume change is 11 gallons per minute.
• BThe drain rate is 11 gallons per minute.
• CThe initial volume in the tank is 11 gallons.
• DThe net rate of volume change is 18 gallons per minute.

In the xy-plane, the graph of the equation $5x + 2y = 40$ 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 $b - a$?

• A$12$
• B$-12$
• C$3$
• D$28$

$mx + 4y = 20$ $6x + ny = 15$ In the given system of equations, m and n are constants. If the system has infinitely many solutions, what is the value of $\displaystyle \frac{m}{n}$?

• A$\displaystyle \frac{16}{5}$
• B$\displaystyle \frac{8}{3}$
• C$\displaystyle \frac{3}{8}$
• D$\displaystyle -\frac{8}{3}$

In the xy-plane, the graph of the equation mx + $5y = 40$ is parallel to the graph of $6x + 15y = 90$, where m is a constant. What is the value of m?

• A$-2$
• B$6$
• C$3$
• D$2$

A company manufactures solar panels. The total cost C, in dollars, to produce p solar panels in a month is given by $C = 12000 + 85p$, where the constant 12000 represents the monthly fixed costs and 85 represents the variable cost per panel. The company sells each panel for $140. The monthly profit P, in dollars, is given by $P = 55p - 12000$. Which statement is the best interpretation of 55 in this context?

• AThe profit per panel sold is $55.
• BThe total variable cost per panel is $55.
• CThe fixed monthly cost is $55.
• DThe profit per panel sold is $85.

In the xy-plane, the graph of the equation $9x - 2y = 36$ has an x-intercept at (c, 0) and a y-intercept at (0, d), where c and d are constants. What is the value of $c - d$?

• A$-14$
• B$11$
• C$22$
• D$7$

In the xy-plane, the graph of the equation tx + $8y = 56$ is parallel to the graph of $3x + 4y = 20$. What is the value of t?

• A$3$
• B$4$
• C$-6$
• D$6$

A moving company charges a flat fee of d dollars plus c dollars per mile to transport furniture. The total cost T, in dollars, for transporting furniture m miles is given by the equation T = cm + d. If it costs 380 dollars to transport furniture 45 miles and 560 dollars to transport furniture 75 miles, what is the value of d?

• A$6$
• B$110$
• C$135$
• D$-110$

In the xy-plane, the graph of the equation $4x - 7y = 56$ passes through the points (a, 0) and (0, b), where a and b are constants. What is the value of $a + b$?

• A$6$
• B$-6$
• C$3$
• D$22$

In the xy-plane, the graph of the equation px + $4y = 20$ is parallel to the graph of $6x + 8y = 15$. What is the value of p?

• A$2$
• B$3$
• C$6$
• D$-3$

In the xy-plane, the graph of the equation $3x - 4y = 24$ 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 $a - b$?

• A$2$
• B$8$
• C$12$
• D$14$

In the xy-plane, the graph of the equation $9x - 6y = 72$ has an x-intercept at (k, 0) and a y-intercept at (0, h), where k and h are constants. What is the value of $k - h$?

• A$-4$
• B$4$
• C$-20$
• D$20$

In the xy-plane, the graph of the equation $2x - 9y = 36$ has an x-intercept at (m, 0) and a y-intercept at (0, n), where m and n are constants. What is the value of $m + n$?

• A$14$
• B$22$
• C$-14$
• D$4$

In the xy-plane, the graph of the equation $9x + 4y = 72$ has an x-intercept at (h, 0) and a y-intercept at (0, k), where h and k are constants. What is the value of $h + k$?

• A$13$
• B$26$
• C$-26$
• D$18$

In the xy-plane, the graph of the equation $8x + 3y = 48$ has an x-intercept at (p, 0) and a y-intercept at (0, q), where p and q are constants. What is the value of $p + q$?

• A$11$
• B$16$
• C$22$
• D$-22$

In the xy-plane, the graph of the equation $6x - 9y = 54$ intersects the y-axis at the point (0, r). What is the value of r?

• A$6$
• B$9$
• C$-6$
• D$-9$

In the xy-plane, the graph of the equation $9x - 4y = k$, where k is a constant, passes through the point (2, 3). What is the x-intercept of this graph?

• A$\displaystyle (\frac{2}{3}, 0)$
• B$\displaystyle (0, -\frac{3}{2})$
• C$\displaystyle (-\frac{2}{3}, 0)$
• D$\displaystyle (\frac{3}{2}, 0)$

In the xy-plane, the graph of $9x + 6y = 72$ 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 $a + b$?

• A$12$
• B$15$
• C$20$
• D$-20$