L1 Mathematical Reasoning
Question Bank
The points $C(1,7)$ and $D(4,13)$ lie on a straight line. Find the equation of the line.
A straight line passes through the points $(-2, 3)$ and $(4, 21)$. Find the equation of the line.
The line shown in the diagram below has a gradient of $-\frac{2}{3}$ and passes through the point $(3, 1)$. Find the equation of the line.
The diagram below shows a straight line graphed on a set of axes. Find the equation of this line.
A straight line has an x-intercept at $(5, 0)$ and a y-intercept at $(0, -15)$. Find the equation of the line and the coordinates of its midpoint.
A tank of water is being emptied through a pipe. The graph shows the volume of water in the tank, $V$ litres, $t$ minutes after the pipe is opened.
(i) Calculate the gradient of the line.
(ii) Explain what the gradient means in this situation.
(iii) Write an equation for $V$ in terms of $t$.
(iv) Use your equation to find when 160 litres are left in the tank.
The graph shows the lines $2x + y = 11$ and $x - y = 1$. The lines cross at the point Q.
Find the coordinates of Q using an algebraic method. Clearly show all steps of your working.
Phone Company A offers a plan for \$30 per month plus \$10 for every gigabyte of data used. Company B offers a plan for \$50 per month plus \$5 for every gigabyte of data used.
By forming two linear equations, calculate the number of gigabytes of data for which the monthly cost of both plans would be identical.
'Zoom Rentals' charges \$70 per day and a one-time administration fee of \$50 to rent a car. 'Coastline Cars' charges \$85 per day with no administration fee.
After how many days would the total rental cost from both companies be exactly the same?
A company offers two different weekly payment structures for its sales team.
Option 1: A base salary of $400 per week, plus a 10% commission on all sales.
Option 2: A base salary of $700 per week, plus a 5% commission on all sales.
What amount of weekly sales would result in the same total earnings from both options?
The diagram below shows a straight line with a gradient of $-\frac{4}{3}$. The line passes through the point shown. Find the y-intercept of the line.
A mobile mechanic charges a \$55 call-out fee and $80 per hour for labour. A workshop charges a flat rate of \$110 per hour with no call-out fee.
By forming two linear equations, determine the number of hours for which the total cost from both providers would be the same.
Aroha and Ben cycle along the same trail from the same starting point. Ben starts one hour after Aroha. The graph shows the distance each has travelled, $d$ km, $t$ hours after Aroha starts.
(i) Use the gradients to find each cyclist's speed. Who is faster?
(ii) Write an equation for each line, giving $d$ in terms of $t$.
(iii) They keep riding at these speeds. Use an algebraic method to find when and where Ben catches up with Aroha.
Line A has the equation $y = 4x - 5$. Line B has the equation $3x + 2y = 22$. Find the coordinates of the point where the two lines intersect.
The diagram below shows two straight lines. Using an algebraic method, find the coordinates of the point where the two lines intersect. You must show how you derived the equations of the lines.
Mere walks from home to a bus stop, waits for the bus, and then rides the bus to school. The graph shows her distance from home.
(i) Calculate Mere's walking speed in km/h.
(ii) Calculate the speed of the bus in km/h.
(iii) Calculate Mere's average speed for the whole journey, in km/h.
(iv) Hemi says the average speed is $\frac{4.8 + 36}{2} = 20.4$ km/h. Explain why Hemi is wrong.
Two ladders are placed in a 12-metre-wide alley, leaning against opposite walls as shown in the diagram. One ladder reaches 8 metres up its wall, and the other reaches 10 metres.
By setting up a coordinate system with the origin (0,0) at the bottom-left corner, model the two ladders as linear equations and find the height, $h$, at which they cross.
