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Cambridge IGCSE0580

Equations

Mathematics 0580 Chapter Notes

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Equations
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In this chapter

  1. 1Constructing expressions and equations
  2. 2Solving linear equations
  3. 3Simultaneous linear equations
  4. 4Changing the subject of a formula
  5. 5Fractional equations
  6. 6Quadratic equations
  7. 7One linear and one non-linear equation

1. Constructing expressions and equations

An expression has no equals sign: 2n + 5. An equation says two things are equal: 2n + 5 = 17.

Read the words one phrase at a time. "More than" means add. "Less than" means subtract. "Times as many" means multiply. Letters written together are multiplied, so 3n means 3 × n.

When two amounts are the same, write one equation with the two amounts either side of the equals sign. Two unknowns need two equations. Consecutive even numbers can be written 2n and 2n + 2.

Examiner insight

“Setting up the initial equation correctly was challenging for most candidates.” “Many candidates were unclear that one equation was needed.”

Worked example 14 marks

Do not use a calculator. Phil has 7y - 52 toy cars. Ann has 78 - 6y toy cars. Phil and Ann have the same number of cars.

(a) Write down an equation to show this information.

(b) Solve your equation and work out the number of cars they each have.

  1. 1

    (a) same number of cars: 7y - 52 = 78 - 6y [1]

  2. 2

    (b) collect the y terms and the numbers: 7y + 6y = 78 + 52 [1]

  3. 3

    13y = 130, so y = 10 [1]

  4. 4

    cars = 7 × 10 - 52 = 18 [1]

2. Solving linear equations

A linear equation has one unknown, to the power 1. Do the same to both sides until the letter is on its own.

Expand brackets first, multiplying every term inside. Then collect the letter terms on one side and the numbers on the other. When a term crosses the equals sign, its sign changes.

The last step divides both sides by the number in front of the letter.

Examiner insight

“Some made errors when expanding the bracket, usually 10x – 7.” “The majority of candidates were able to expand one or both brackets correctly although not all were able to rearrange their resulting equation into a correct ax = b form.”

Worked example 13 marks

Solve.

5(2x - 7) = 79

  1. 1

    expand the bracket: 10x - 35 = 79 [1]

  2. 2

    collect: 10x = 79 + 35 = 114 [1]

  3. 3

    divide by 10: x = 11.4 [1]

3. Simultaneous linear equations

Two equations in x and y are solved together. The usual method is elimination.

Multiply one or both equations so that one letter has the same number in front of it in both. If the signs of that letter are different, add the equations. If the signs are the same, subtract them. Multiply every term of an equation, including the right-hand side.

Solve for the first letter. Substitute it into either original equation to find the other. Check in the equation you did not use.

Examiner insight

“Some candidates made errors when deciding whether to add or subtract equations, particularly when working with negative signs.”

Worked example 14 marks

Do not use a calculator. Solve the simultaneous equations.

-5x - 7y = 1

6x + 8y = -2

  1. 1

    multiply the first by 8 and the second by 7: -40x - 56y = 8 and 42x + 56y = -14 [1]

  2. 2

    the y terms have opposite signs, so add: 2x = -6 [1]

  3. 3

    x = -3 [1]

  4. 4

    substitute in 6x + 8y = -2: -18 + 8y = -2, 8y = 16, so y = 2 [1]

4. Changing the subject of a formula

The subject is the letter on its own on one side, such as v in v = u + at. To change the subject, undo the operations around the new subject in reverse order, doing the same to both sides.

When the new subject appears twice, collect every term containing it on one side. Then factorise it out and divide by the bracket.

When the subject has a power or root, isolate that term first. Then take the root, or square, the whole of the other side.

Examiner insight

“Many candidates knew to start by collecting terms with p on one side of the formula and terms without p on the other.”

Worked example 13 marks

Make p the subject of ap + 4 = 2p + 5a.

  1. 1

    collect the p terms on one side: ap - 2p = 5a - 4 [1]

  2. 2

    factorise: p(a - 2) = 5a - 4 [1]

  3. 3

    divide by (a - 2): p = (5a - 4)/(a - 2) [1]

5. Fractional equations

A fractional equation has the unknown in a numerator or denominator, such as x/(2x + 1) = 4 or (4x - 3)/5 = (7 - 2x)/3.

Cross multiply to remove the fractions. When two fractions are added, multiply every term by each denominator. Put brackets round any numerator with more than one term.

Then expand the brackets and solve as an ordinary equation. Take care when collecting terms across the equals sign.

Examiner insight

“Some candidates made sign errors when collecting the x terms to one side after eliminating the fractions and expanding brackets.”

Worked example 13 marks

Solve.

(4x - 3)/5 = (7 - 2x)/3

  1. 1

    cross multiply: 3(4x - 3) = 5(7 - 2x) [1]

  2. 2

    expand: 12x - 9 = 35 - 10x [1]

  3. 3

    22x = 44, so x = 2 [1]

6. Quadratic equations

Write the equation as ax² + bx + c = 0 first. Then choose a method.

Factorise into two brackets and set each to zero. The quadratic formula works for every quadratic. Completing the square writes the expression as (x + p)² + q. Halve the number in front of x to find p.

Put negative numbers in brackets when you substitute: (-5)² = 25. Give surd answers when asked for exact form.

x = (-b ± √(b² - 4ac))/2a

x² + bx + c = (x + b/2)² - b²/4 + c

Examiner insight

“A few tried to factorise but most candidates did use the ‘quadratic formula’, the most common error being the fraction line being too short and not including the ‘–b’ term.”

Worked example 13 marks

Do not use a calculator. Solve by factorisation x² + 16x - 36 = 0 to find the length x cm.

  1. 1

    two numbers with product -36 and sum 16 are 18 and -2: (x + 18)(x - 2) = 0 [2]

  2. 2

    x = -18 or x = 2. A length is positive, so x = 2 [1]

7. One linear and one non-linear equation

A line and a curve meet where both equations are true. Make one letter the subject of the linear equation. Substitute it into the other equation.

This gives a quadratic in one letter. Solve it, then put each answer into the linear equation to find the matching value of the other letter. Give each pair of values as a point.

Examiner insight

“Some used the quadratic equation of the curve to find the corresponding y values instead of using the simpler equation of the line.”

Worked example 15 marks

Do not use a calculator. The straight line y = 7 - 4x intersects the curve y = 4 + 3x - 2x² at two points. Find the coordinates of these two points.

  1. 1

    equate the two expressions for y: 7 - 4x = 4 + 3x - 2x² [1]

  2. 2

    rearrange: 2x² - 7x + 3 = 0 [1]

  3. 3

    factorise: (2x - 1)(x - 3) = 0 [1]

  4. 4

    x = 0.5 or x = 3 [1]

  5. 5

    use the line y = 7 - 4x: the points are (0.5, 5) and (3, -5) [1]

Recap

  • An expression has no equals sign. An equation does. Two equal amounts give one equation.
  • Expand every term in a bracket. A term that crosses the equals sign changes sign.
  • Elimination: equal numbers in front of one letter. Different signs, add. Same signs, subtract.
  • Find one letter, substitute to find the other, then check in the other equation.
  • Changing the subject: collect, factorise, divide. Root or square the whole side.
  • Fractions: cross multiply. Quadratics: factorise or formula. Line and curve: substitute.

End-of-chapter exercise

Test yourself on the whole chapter. Work through these before moving on.

  1. Do not use a calculator. A box holds n sweets. Sam has 3 boxes and 5 loose sweets. (a) Write an expression, in terms of n, for the number of sweets Sam has. (b) Sam has 38 sweets in total. Write an equation and find n.3 marks
  2. Use a calculator. Solve 5 - 2x = 3(x + 7).3 marks
  3. Do not use a calculator. Solve the simultaneous equations. 3x + 2y = 7 5x - 2y = 13 marks
  4. Use a calculator. Two adult tickets and three child tickets cost 41 dollars. One adult ticket and four child tickets cost 38 dollars. Work out the cost of one adult ticket.4 marks
  5. Do not use a calculator. Make w the subject of P = 2(l + w).2 marks
  6. Do not use a calculator. Solve x/(2x + 1) = 4.3 marks
  7. Use a calculator. Solve the simultaneous equations. y = x + 2 x² + y² = 205 marks
  8. Do not use a calculator. Make r the subject of A = 3r² + 5.3 marks
  9. Use a calculator. Ali walks the first part of a route at x km / h and the second part at (x + 2) km / h. The first part takes 6/x hours and the second part takes 10/(x + 2) hours. The whole route takes 3 hours. (a) Show that 3x² - 10x - 12 = 0. (b) Find the value of x. Give your answer correct to 2 decimal places.6 marks
  10. Do not use a calculator. A rectangle has length x cm and width y cm. Its perimeter is 20 cm and its area is 23 cm². The length is greater than the width. (a) Show that x² - 10x + 23 = 0. (b) Solve this equation by completing the square. Find the exact length and the exact width.6 marks

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