Simultaneous equations

Kai plays an online game. Kai collects diamonds and rubies. Each diamond and each ruby is worth a fixed number of points.

In game 1, Kai collects 4 diamonds and 3 rubies and scores 47 points. In game 2, Kai collects 3 diamonds and 5 rubies and scores 49 points.

Work out the number of points for a diamond and the number of points for a ruby. You must show your working.

diamond =........ points

Function A and function B are shown below.

When the input for both machines is 11, the output of Function A is equal to the output of Function B. When the input for both machines is 29, the output of Function A is twice the output of Function B.

Find the value of a and the value of b.

The graph of y =-3x intersects the graph of x² + y² = 70 at two points.

Work out the exact coordinates of the two points. You must show your working.

(........,........) and (........,........)

Solve the simultaneous equations. 4 x - y = 24 2 x + 3 y = 26

Four numbers are written, in ascending order, as algebraic expressions.

a a + b a + 2 b 3 a - b

The mean of these four numbers is 27. The range of these four numbers is 24.

Find the value of a and the value of b. You must show your working.

Work out the coordinates of the intersection of the graphs of y = 5 - 2x and y = 3x². You must show your working.

(........,........) and (........,........)

2 kg of carrots and 5 kg of potatoes cost £6.36. 3 kg of carrots and 2 kg of potatoes cost £5.25.

Find the cost of 1 kg of carrots and the cost of 1 kg of potatoes. You must show your working.

1 kg of carrots cost £........

1 kg of potatoes cost £........

2 kg of carrots and 5 kg of potatoes cost £6.36. 3 kg of carrots and 2 kg of potatoes cost £5.25.

Find the cost of 1 kg of carrots and the cost of 1 kg of potatoes. You must show your working.

1 kg of carrots cost £........

1 kg of potatoes cost £........

The x-coordinates of the intersections of the graphs of y = x² + ax + 7 and y = 3x + b are the solutions to the equation x² + 8x + 13 = 0.

Find the value of a and the value of b.

A six-sided numbered spinner is thrown 50 times. The score for each throw is recorded. Some of the results are shown in the table.

An 8 was thrown f times. An unknown number on the spinner is represented by n.

The mean score of the 50 throws is 5.5.

Find the value of f and the value of n.

A student draws two different regular polygons. The exterior angle of one polygon is p °. The exterior angle of the other polygon is q °.

The sum of p and q is 112°. The difference between p and q is 32°.

Find the number of sides of each polygon. You must show your working.

........ sides and........ sides

A six-sided numbered spinner is thrown 50 times. The score for each throw is recorded. Some of the results are shown in the table.

An 8 was thrown f times. An unknown number on the spinner is represented by n.

The mean score of the 50 throws is 5.5.

Find the value of f and the value of n.

A student draws two different regular polygons. The exterior angle of one polygon is p °. The exterior angle of the other polygon is q °.

The sum of p and q is 112°. The difference between p and q is 32°.

Find the number of sides of each polygon. You must show your working.

........ sides and........ sides

Describe fully the graph of x² + y² = 20.

The graph of y = 3x + 10 intersects the graph of x² + y² = 20 at two points.

Use an algebraic method to work out the coordinates of the two points. You must show your working.

(b) (........,........) and (........,........)

A theatre has an adult price and a child price for their shows.

A group of 4 adults and 5 children paid a total of £136. A group of 3 adults and 2 children paid a total of £81.

Work out the price for one adult and the price for one child. You must show your working.

Price for one adult £........

Price for one child £........

The next term in a Fibonacci sequence is found by adding together the two previous terms.

The first and second terms of a particular Fibonacci sequence are x and y.

Show that the fourth term of the sequence can be written as x + 2 y.

The fourth term of the same Fibonacci sequence is 7. The seventh term of the sequence is 31.

Work out the value of x and the value of y. You must show your working.

Here are the first four terms of a sequence.

Write an expression for the n th term.

Here are the first four terms of a quadratic sequence.

– 1 5 13 23

The nth term is n² + bn + c.

Find the value of b and the value of c.

Solve algebraically. x² + y² = 18 y = x - 6

Solve the simultaneous equations.

3 x + y = 11 x + y = 3

Kai buys 5 drinks and 3 cakes for £16.35. Azmi buys 2 drinks and 6 cakes for £14.70.

Assume that each drink costs the same and that each cake costs the same.

Calculate the cost of one drink and the cost of one cake. You must show your working.

Cost of one drink £........

Cost of one cake £........

Solve algebraically.

y = x + 3 (x - 3)² + y² = 50

You must show your working.

x² - 2y = 5 and 4y + z = 7.

Write z in terms of x. Give your answer in its simplest form.

Solve the simultaneous equations.

2 x + 3 y = 10 3 x + 5 y = 17

Solve.

x² + y² = 34 y = x + 2

Show your working.

Li has t toy bricks. She only has red bricks and blue bricks.

Li picks two bricks, one after the other. If the first brick she picks is red, the probability that the second brick is red is If the first brick she picks is blue, the probability that the second brick is red is Calculate the value of t.

Solve the simultaneous equations.

2 x - y = 7 2 x + y = 5

The diagrams show the price paid by two groups of people visiting a funfair.

Assume each adult pays the same price and each child pays the same price.

Find the price for an adult and the price for a child.

James and Elizabeth buy some clothes.

James buys 5 shirts and 4 jumpers. He pays £163. Elizabeth buys 3 shirts and 2 jumpers. She pays £89.

Assume that each shirt has the same cost and that each jumper has the same cost.

Work out the cost of one shirt and the cost of one jumper. You must show your working.

Cost of one shirt £........

Cost of one jumper £........

Here are the first four terms of a sequence.

- 1 4 9 14

Write an expression for the n th term of this sequence.

The n th term of another sequence is given by

The third term is 9 and the sixth term is 126.

Find the value of a and the value of b.

Find the exact coordinates of the two intersections of the line y = 2x and the circle x² + y² = 30

(........,........) and (........,........)

The dimensions, in centimetres, of this rectangle are shown as algebraic expressions.

3 x + 5 y + 4

Work out the length and width of the rectangle.

Four identical trapeziums are placed on a coordinate grid as shown.

Write down algebraic expressions for the coordinates of point P.

The coordinates of point Q are (16, 13).

Work out the value of a and the value of b.

The dimensions, in centimetres, of this rectangle are shown as algebraic expressions.

5 x – y – 8
Not to scale

Work out the length and width of the rectangle.

A straight line passes through the point (0, 6) and is perpendicular to y = 4 x - 5.

Find the equation of this line, giving your answer in the form y = mx + c.

Work out the coordinates of the intersection of the graphs of y = 4x - 5 and y = x² - 17.

Two congruent, isosceles triangles are joined, as shown, to form a parallelogram. The largest angle of the parallelogram is 110°.

Write two equations. Solve them to find the value of a and the value of b.

Solve these simultaneous equations algebraically.

A sequence is defined using this term-to-term rule.

Another sequence is defined using this term-to-term rule,

where k and r are constants.

Given that u₂ = 41, u₃ = 206 and u₄ = 1031, find the value of k and the value of r.

Each week Dan drives two routes, route X and route Y.

One week he drives route X three times and route Y twice. He drives a total of 134 miles that week.

Another week he drives route X twice and route Y five times. He drives a total of 203 miles that week.

Find the length of each route.

(a) route X =........ miles

State an assumption that has been made in answering part (a).

Each week Dan drives two routes, route X and route Y.

One week he drives route X three times and route Y twice. He drives a total of 134 miles that week.

Another week he drives route X twice and route Y five times. He drives a total of 203 miles that week.

Find the length of each route.

(a) route X =........ miles

State an assumption that has been made in answering part (a).