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

Collecting Like Terms
1.
 * Here we only have $$x$$ terms, so we can add them together:
 * $$x + x = 2x$$

2.
 * We have a similar situation to problem 1, but this time with $$x^2$$ terms. We can combine them by adding their coefficients:
 * $$x^2 + 3x^2 = (1 + 3)x^2 = 4x^2$$

3.
 * Here we have three different terms in the mix ($$x$$, $$x^2$$ and $$x^3$$) so we need to be a little more careful. We can only combine coefficients if they represent the same term. Separate the expression into terms of $$x$$, $$x^2$$ and $$x^3$$ as follows:


 * $$(3x + 2x) + (2x^2 -2x^2) + 3x^3$$


 * Now combine the coefficients of the like terms:


 * $$(3 + 2)x + (2 - 2)x^2 + 3x^3 = 5x + 3x^3$$

4.
 * This is another case with three distinct terms ($$zy$$, $$z$$ and $$y$$). Combine the coefficients of like terms:



\begin{align} zy + 2zy + 2z + 2y &= (1 + 2)zy + 2z + 2y \\&= 3zy + 2z + 2y \end{align} $$

5. $$-x^2 + 4x^2y - 4xy^2 + 7xy$$

Multiplication

 * 1) $$4x^2$$
 * 2) $$18x^2y^2$$
 * 3) $$36ab^2xz$$
 * 4) $$60x^5y^4z^2$$
 * 5) $${\sqrt{x^5}}$$

Fractions

 * 1) $$x$$
 * 2) $$\frac{7x}{12}$$
 * 3) $$\frac{16xy}{15}$$
 * 4) $$2x - y + z$$
 * 5) $$\frac{x^2+y^2}{xy}$$

Changing the Subject of an Equation

 * 1) Solve for x.

$$x = \frac{y}{2}$$
 * 1) Solve for z.

$$z = \frac{x - 8}{3}$$
 * 1) Solve for y.

$$y = b^2$$
 * 1) Solve for x.

$$x = \sqrt {y + 9}$$
 * 1) Solve for b.

$$b = \frac{6y + 7z}{6}$$

Solving Quadratic Equations
Find the Roots of:
 * 1) $$x = -2\ or\ 3$$
 * 2) $$x = 7\ or\ \frac{3}{2}$$
 * 3) $$x = 2\ or\ 3$$
 * 4) $$x = -1\ or\ 0$$
 * 5) $$x = -3\ or\ 4$$

Simultaneous Equations

 * 1) $$a = 4, s = 2$$
 * 2) $$c = \frac{2}{5}, d = 1$$
 * 3) $$b = 7, g = 12$$

Equations/Solutions