Algebra I: Linear Equations, Inequalities, Systems, and Graphs

A linear equation has a variable only to the first power. Solve it by using inverse operations to isolate the variable. Whatever operation is performed on one side of the equation must also be performed on the other.

Example: Solve $3(x-2)+5=20$.

3x-6+5&=20\\ 3x-1&=20\\ 3x&=21\\ x&=7 \end{aligned}$$ Check: $3(7-2)+5=15+5=20$, so $x=7$ is correct.

Solve a linear inequality much like an equation. The key extra rule is:

When multiplying or dividing both sides by a negative number, reverse the inequality sign.

Example: Solve $-2x+4\ge 10$.

-2x&\ge 6\\ x&\le -3 \end{aligned}$$ The sign reverses because both sides were divided by $-2$. On a number line, $x\le -3$ is shown with a closed circle at $-3$ and shading to the left.

A system of equations consists of two or more equations. Its solution is a point that makes every equation true. On a graph, it is the intersection point of the lines.

Two common methods are substitution and elimination.

Example using elimination:

x+y&=9\\ x-y&=3 \end{aligned}$$ Add the equations to eliminate $y$: $$2x=12 \quad\Rightarrow\quad x=6$$ Substitute into $x+y=9$: $$6+y=9 \quad\Rightarrow\quad y=3$$ So the solution is $(6,3)$. A system can have one solution (intersecting lines), no solution (parallel lines), or infinitely many solutions (the same line).

A function assigns exactly one output to each input. A linear function is often written in slope-intercept form:

$y=mx+b$

Here, $m$ is the slope, or rate of change, and $b$ is the $y$-intercept, where the line crosses the $y$-axis.

Example: Graph $y=-2x+3$.

• The $y$-intercept is $3$, so plot $(0,3)$.
• The slope is $-2=-\frac{2}{1}$. From $(0,3)$, move down $2$ and right $1$ to get $(1,1)$.
• Draw the line through these points.

A positive slope rises from left to right; a negative slope falls. To find slope from two points, use

$m=\frac{y_2-y_1}{x_2-x_1}.$