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

Manipulating Equations

Collecting Like Terms

Simplify the following:

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

Multiplication

Simplify the following:

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

Fractions

Simplify the following:

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

Changing the Subject of an Equation

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

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

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:

$b+5=g$

$8b+9g=164$

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