Linear Equations, Systems & Modelling

Practice Questions

Level 1: Easy (Foundational)

1. A line has equation $2x - 3y = 12$. What is its $y$-intercept?
   A) $-4$
   B) $4$
   C) $-6$
   D) $6$
2. If $f(x)=5x-7$, what is the value of $x$ for which $f(x)=13$?
   A) $3$
   B) $4$
   C) $-4$
   D) $20$
3. A taxi charges a fixed $\$3$ pickup fee plus $\$1.50$ per kilometre. Determine the exact cost, in dollars, of an $8$ km trip. (SPR)

Level 2: Medium (Standard)

1. Solve the system $3x + 2y = 12$, $x - y = 1$. Which is the value of $y$?
   A) $\tfrac{9}{5}$
   B) $\tfrac{7}{5}$
   C) $2$
   D) $3$
2. For what value of $k$ does the system $x + ky = 3$, $2x + 4y = 6$ have infinitely many solutions?
   A) $1$
   B) $2$
   C) $3$
   D) $4$
3. The line $y = 2x + b$ passes through $(3, 11)$. Determine the exact value of $b$. (SPR)
4. A rope $L$ metres long is cut into two pieces so the longer piece is $2$ m more than twice the shorter. If the shorter piece is $s$ m, which equation is correct?
   A) $L = 3s + 2$
   B) $L = 3s - 2$
   C) $L = 2s + 2$
   D) $L = s + 2$

Level 3: Hard (Advanced)

1. The system $\begin{cases}3x - 2y = 8 \\ 6x + ky = 5\end{cases}$ has no solution. Determine the value of $k$.
   A) $-4$
   B) $4$
   C) $-8$
   D) $8$
2. Line $\ell$ has slope $m$ and passes through $(2,5)$ and $(6,-3)$. Determine the exact value of $m$. (SPR)
3. A savings account starts at $\$400$ and grows by a fixed $\$25$ each month. After how many months $t$ does the balance first exceed $\$1000$?
   A) $23$
   B) $24$
   C) $25$
   D) $26$