Simultaneous Equations, Inequalities and Regions

In short: Simultaneous equations are two equations that are both true at once, and their solution is the point where their graphs cross. Solve them by substitution or elimination. Solve inequalities like equations, but flip the sign when you multiply or divide by a negative number.

Starting point: Years 9 and 10

Linear inequalities

An inequality uses $<$, $>$, $\le$ or $\ge$. Solve it like an equation, with one extra rule:

$$3-2x>9\Rightarrow -2x>6\Rightarrow x<-3$$

On a number line, an open circle means the end is not included ($<$ or $>$). A closed circle means it is included ($\le$ or $\ge$).

Simultaneous equations

Two equations with two unknowns. The solution is the point that makes both true, which is where their graphs cross.

Substitution works well when one equation already says $y=\dots$ or $x=\dots$.

Elimination works well when the equations line up. Add or subtract them to remove one letter. You may need to multiply first.

Word problems

1. Name the two unknowns.
2. Write two equations from the facts.
3. Solve and answer in words.

Regions on the number plane

The line $y=x+1$ splits the plane into two half-planes. $y>x+1$ is everything above the line, and $y<x+1$ is everything below it.

• Draw the boundary dashed for $<$ or $>$ (not included).
• Draw it solid for $\le$ or $\ge$ (included).
• Test a point such as $(0,0)$ to see which side to shade.

Quick summary

• Solve inequalities like equations, but flip the sign when multiplying or dividing by a negative.
• Simultaneous equations: the point that makes both true, where the graphs cross.
• Substitution when one letter is already alone. Elimination when the equations line up.
• Regions: dashed boundary for $<$ and $>$, solid for $\le$ and $\ge$. Test a point to choose the side.
• A line and a parabola meet 0, 1 or 2 times. Use the discriminant.

Quiz