Simultaneous Equations (AQA A Level Maths: Pure)

Use elimination to solve the simultaneous equations

Use substitution to solve the simultaneous equations

By eliminating y from the equations

show that .

Hence solve the simultaneous equations

By eliminating y from the equations

show that .

Hence solve the simultaneous equations

By eliminating y from the equations

show that .

Hence solve the simultaneous equations

giving x and y in the form , where a and b are integers.

are simultaneous equations, where k is a constant.

Show that .

Find an expression for y in terms of the constant k.

Given that , find the value of k.

are simultaneous equations, where k is a constant.

By eliminating y from the equations show that .

By considering the discriminant of find the value of k for which the simultaneous equations have only one solution.

Find the solution to the simultaneous equations for the value of k that you found in part (b).

You are asked to advise a client on which parcel delivery service to use to deliver parcels of differing sizes.  Linear Deliveries Inc. charges a flat rate of £2.25 per parcel, plus 40p times the mass of the parcel in kilograms.  Square Deal Delivery Solutions charges a flat rate of £4 per parcel, plus 1p times the square of the parcel’s mass in kilograms.  Under what circumstances would you advise your client to use each of the two delivery services?  Be sure to show clear mathematical justifications for your answer.

Use elimination to solve the simultaneous equations

Use substitution to solve the simultaneous equations

Solve the simultaneous equations

By eliminating y  from the equations

show that .

Hence solve the simultaneous equations

By eliminating y  from the equations

show that .

Hence solve the simultaneous equations

giving  and  in the form , where and  are rational numbers.

are simultaneous equations, where  is a constant.

Solve the equations for  and , giving your answer for  in terms of the constant .

For what value of the constant will the values of  and  in the solution be equal?

are simultaneous equations, where  is a constant.

Given that the simultaneous equations have exactly one solution, find the value of the constant .

Find the solution to the simultaneous equations for the value of  that you found in part (a).

A firework is launched inside a large shed with a sloping roof.  In relation to the horizontal distance from the point it was launched, the height of the firework, m, can be modelled by the quadratic equation

The sloping roof of the shed can be modelled with the equation

Determine whether, according to the model, the firework will hit the roof of the shed before escaping out the open end of the shed on the right of the diagram.

Use elimination to solve the simultaneous equations

Use substitution to solve the simultaneous equations

Solve the simultaneous equations

Solve the simultaneous equations

By eliminating  from the equations

 show that .

Hence solve the simultaneous equations

giving  and  in the form , where  and  are rational numbers and  is a prime number.

are simultaneous equations, where  is a constant.

Solve the equations for  and , giving your answer for in terms of the constant .

For what value of the constant  do the equations not have a solution?

are simultaneous equations, where  is a constant.

Find the respective sets of values for k for which the simultaneous equations have one, two, and no solutions.

Given that the simultaneous equations have exactly one solution, find all possible pairs  that might correspond to that solution. Give all your values for  and  in the form , where  is a rational number.

The goal in a video game is to have a unicorn leap as far as possible in a horizontal direction without being destroyed by the death ray that is being fired overhead.  You hack into the game code and find that the height of the unicorn, , is being modelled in relation to the horizontal distance from the point it jumps by the quadratic equation  , where  is a parameter that can be controlled by the player’s actions, and  is the horizontal distance in metres.  You also find that the path of the death ray is being modelled by the equation . 

The value of can never be less than zero, and if the path of the unicorn crosses or touches the path of the death ray, the unicorn is considered to have been destroyed.

Ignoring the problem of the death ray, explain why the parameter  represents the horizontal distance leapt by the unicorn.

Your friend’s personal best in the game is a leap of 21.5 m without the unicorn being destroyed. He is determined to keep playing until his unicorn has leapt 22 m safely.  Determine whether or not your friend has a chance of reaching this goal.