A-level Mathematics/OCR/C1/Equations/Problems

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Manipulating Equations[edit]

Collecting Like Terms[edit]

  1. x + x
  2. x^2 + 3x^2
  3. 3x + 2x^2 + 2x -2x^2 + 3x^3
  4. zy + 2zy + 2z + 2y
  5. 8x^2 + 7xy + x^2 - 10x^2 + 4x^2y - 4xy^2

Multiplication[edit]

  1. 2x \times 2x
  2. 6xy \times 3xy
  3. 6zb \times 3x \times 2ab
  4. 3x^2 \times 4xy^2 \times 5x^2y^2z^2
  5. x^2 \sqrt {x}

Fractions[edit]

  1. \frac{x}{2} + \frac{x}{2}
  2. \frac{x}{3} + \frac{x}{4}
  3. \frac{3xy}{15} - \frac{xy}{3} + \frac{6xy}{5}
  4. \frac{4x}{2} - \frac{4y}{4} + \frac{8z}{8}
  5. \frac{x}{y} + \frac{y}{x}

Solving Equations[edit]

Changing the Subject of an Equation[edit]

  1. Solve for x.

    y = 2x

  2. Solve for z.

    x = 3z + 8

  3. Solve for y.

    b = \sqrt{y}

  4. Solve for x.

    y = x^2 - 9

  5. Solve for b.

    y = \frac{6b - 7z}{6}

Solving Quadratic Equations[edit]

Find the Roots of:

  1. x^2 - x -6 = 0
  2. 2x^2 - 17x + 21 = 0
  3. x^2 - 5x + 6 = 0
  4. x^2 + x = 0
  5. -x^2 + x + 12 = 0

Simultaneous Equations[edit]

Example 1

At a record store, 2 albums and 1 single costs £10. 1 album and 2 singles cost £8. Find the cost of an album and the cost of a single.

Taking an album as a and a single as s, the two equations would be:

2a+s=10

a+2s=8

You can now solve the equations and find the individual costs.

Example 2

Tom has a budget of £10 to spend on party food. He can buy 5 packets of crisps and 8 bottles of drink, or he can buy 10 packets of crisps and 6 bottles of drink.

Taking a packet of crisps as c and a bottle of drink as d, the two equations would be:

5c+8d=10

10c+6d=10

Now you can solve the equations to find the cost of each item.

Example 3

At a sweetshop, a gobstopper costs 5p more than a gummi bear. 8 gummi bears and nine gobstoppers cost £1.64.

Taking a gobstopper as g and a gummi bear as b, the two equations would be:

b+5=g

8b+9g=164