x y = 1 x + 2y = 7 y = x + 1 y = 2x 4 Solve these simultaneous equations. 2x + 3y = 8 2x + 3y = 6 4x y = 6 3 = x + 2y Guided practice worksheet

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1 7. Solving simultaneous equations Questions are targeted at the grades indicated Solve these simultaneous equations. + = = = + = 7 = = B + = + = 8 = + = = = = = 8 + = = = = 6 9

2 7. Solving simultaneous equations + = = = = B + 7 = + + = 8 + = = + = = + + = 8 = 9B

3 7. Setting up equations in two unknowns The sum of two numers is 7 and twice the larger numer eceeds three times the smaller numer. Find the value of each of the numers. Questions are targeted at the grades indicated Si pencils and three ruers cost 6. Five pencils and two ruers cost.9 Work out the cost of pencils and ruers.... The diagram shows an isosceles triangle. ll the angles are measured in degrees. p q... nut and a olt have a mass of 98 g. The mass of olts and nuts is 6 g. Find the mass of a nut and the mass of a olt. p + q p + q Find the size of each angle

4 7. Setting up equations in two unknowns The average of two mied numers is. One of the numers is more than the other. Find the value of each numer. 6 Strawerr jam costs p per pot more than a pot of marmalade. Two pots of strawerr jam and three pots of marmalade cost 7.6 Work out the cost of a pot of marmalade In m office, I have two-drawer filing cainets and three-drawer filing cainets. Each drawer of a two-drawer filing cainet holds 6 files. Each drawer of a three-drawer filing cainet holds files. I have a total of 8 files and all m filing cainets are full. a Show that + = 8... I have more three-drawer filing cainets than two-drawer filing cainets in m office. Work out the numer of each tpe of filing cainet B

5 7. Using graphs to solve simultaneous equations Questions are targeted at the grades indicated B B C a The line has equation + = Write down the equation of each of the lines B and C Write down the pair of simultaneous equations that have a solution. i =, =... ii =, =... iii =, =... For each of these pairs of simultaneous equations, draw two linear graphs on the same grid and use them to solve the simultaneous equations. a + = =

6 7. Using graphs to solve simultaneous equations = + = 7 B c = + + = d + = = e = = B

7 7. Solving quadratic equations factorisation Questions are targeted at the grades indicated Hint If the product of epressions is zero, then one of them must equal zero. Solve a ( + 8) = ( )( + ) = c ( )( ) = B Solve a + = = c 6 =... Solve... a 8 + = + + = c + = d = e = f + + = Solve a 9 = =

8 7. Completing the square Epand and simplif. a ( + ) = ( + )( + ) =... ( + ) =... =... c ( + 8) =... =... d ( ) =... =... C Use our answers to Question to complete the following. a + = ( + )... + = ( + )... c + 6 = ( + 8)... d = ( )... a Epand and simplif ( + 6) = (n + p) + q Find the values of p and q. p =... q =... a Epand and simplif ( 9) = ( c) d Find the values of c and d. c =... d =... a Write + + in the form ( + e) + f.... State the values of e and f. e =... f =... 6 Write + + in the form ( + g) + h.... 7

9 7.6 Solving quadratic equations completing the square Questions are targeted at the grades indicated Here is a diagram for + B completing the diagram for the part of the square that equates to +, epress + in the form ( + p) + q....? ²?......?? Using the method of completing the square, solve these quadratic equations, giving our solutions in surd form. a = = c + = d 6 8 = e = f =

10 7.6 Solving quadratic equations completing the square Using the method of completing the square, solve these quadratic equations, giving our solutions correct to decimal places. a = + = c + = d = e 6 6 = f 9 = B

11 7.7 Solving quadratic equations using the formula Questions are targeted at the grades indicated For each of these quadratic equations, a work out the value of ac, and Hint The whole of the numerator is divided a. use the quadratic formula = ± ac, to solve them. a Give our solutions correct to significant figures. + + = = + = a... a... a = = = a... a... a = = 9 + = a... a... a = + = + 6 = a... a... a

12 7.7 Solving quadratic equations using the formula Which of the following quadratic equations have a two solutions, one solution, c no solution? + + = = = = = =... B

13 7.8 Solving algeraic fraction equations leading to quadratic equations Questions are targeted at the grades indicated Solve these fraction equations. a + = = * c + = d + = e = + f + =

14 7.8 Solving algeraic fraction equations leading to quadratic equations Show that the equation = can e written as 7 = and hence solve = 7 *... Solve these fraction equations. Give our solutions correct to significant figures. a + + = = c = d + = e = f + = B

15 7.9 Setting up and solving quadratic equations Questions are targeted at the grades indicated The formula S n( n + ) gives the sum S of the numers,,, etc up to n. Find the value of n if S = 6. = *... Find the numer such that twice its square is more than times the original numer.... Find the ages of a lad and her daughter if the lad was ears old when her daughter was orn, and the product of their ages is now The area of an oil painting is cm. The width is cm less than the length. Find the length of the picture. Give our answer correct to decimal places....

16 7.9 Setting up and solving quadratic equations garage is 9 metres long. The area of the rectangular walls is m. The height is m less than the width. Find the height of the garage. *... 6 rectangular lawn measures m m. The lawn is surrounded a path of uniform width. If the area of the path is 6 m, find its width The perimeter of a right-angled triangle is 6 cm. The length of the hpotenuse is cm. Calculate the lengths of the other two sides Michael is training for a long-distance ccle race. One da he ccles for hours in the morning, travelling a distance of 8 km. In the afternoon, he ccles for hour more travelling the same distance, ut his average speed is km/h slower than in the morning. Calculate the numer of hours that Michael ccles in the afternoon.... B

17 7. Constructing graphs of simple loci Use squared paper for our answers. a Draw the locus of points units from the i -ais ii -ais. Write down the equations for each locus. Questions are targeted at the grades indicated C i... ii... a Use squared paper to draw each of the following lines. i = 6 ii = 6 iii = iv = B Use dotted lines to draw the locus of points units from each of the aove lines. c Write down the equations of each locus. i... ii... iii... iv... 7

18 7. Constructing graphs of simple loci The graph shows the line with equation + =. B a + = The dotted lines are the locus of points units from the line. a Use Pthagoras theorem to calculate the distance a Find the equations of the locus a Use the graph opposite to draw a circle of radius 6 with the origin (, ) as centre. On the same aes, draw the line with equation + =. c Calculate the equations of the two tangents to the circle that are parallel to + = d Draw the tangents. Hint The angle etween a tangent and a radius of a circle is B

19 7. Constructing graphs of simple loci a Use graph paper to draw the locus of points 7 units from the origin Hint Draw a circle with centre at the origin Complete the equation of the locus: + =... c Draw the tangent to the circle at i (7, ) ii (, 7) d Write down the equation of each tangent. i... ii... 6 Find the equations of the locus of points 6 units from the line + =. * C

20 7. Solving simultaneous equations when one is linear and the other is quadratic Questions are targeted at the grades indicated For each of these pairs of simultaneous equations, a draw a quadratic graph and a linear graph on the same grid and use them to solve the simultaneous equations solve them using an algeraic method. Give our answers correct to decimal places where appropriate. You must show all of our working. + = 6 = a = = = 7 = + a = =

21 7. Solving simultaneous equations when one is linear and the other is quadratic = = a = = = + 8 = a = = B

22 7. Solving simultaneous equations when one is linear and one is a circle On graph paper, draw the graph of the circle with equation + =. On the same aes, draw the straight line with equation =. Hence find estimates of the solutions of the simultaneous equations + = and =. Questions are targeted at the grades indicated * Draw suitale graphs to find estimates of the solutions of the simultaneous equations + = 9 and =. Draw suitale graphs to find estimates of the solutions of the simultaneous equations + = and + =

23 7. Solving simultaneous equations when one is linear and one is a circle Solve algeraicall these simultaneous equations. Give our answers correct to significant figures where appropriate. a + = + = c + = + = 7 = = + * Solve algeraicall these simultaneous equations. Give our answers correct to significant figures. a + = + = c + = 7 = + = = B

Section A Plotting Straight Line Graphs Grade D / C

Name: Teacher Assessment Section A Plotting Straight Line Graphs Grade D / C 1. (a) Complete the table of values for = 3x + x 0 1 3 5 10 16 19 (b) On the grid draw the graph of = 3x + for values of x from