Chapter 1: Linear Functions & Systems of Equations

Linear functions are the starting point of Algebra 2 and form the foundation for every topic that follows. In this chapter, we develop a thorough understanding of slope, the three standard forms of a linear equation, and the relationships between parallel and perpendicular lines. We then move to systems of equations—first in two variables, then in three—and finish with systems of linear inequalities and an introduction to linear programming. Every concept is reinforced with worked examples, interactive Desmos graphs, and practice problems so you build both computational skill and geometric intuition.

1.1 Slope & Linear Equations

The concept of slope captures how steep a line is and in which direction it tilts. Slope is the single most important number associated with a linear function: it tells you the rate at which the output changes per unit change in the input. In applications, slope represents speed, cost per item, temperature change per hour, and countless other rates.

The beauty of slope is that it is constant everywhere on a given line. No matter which two points you select, the ratio $\frac{\Delta y}{\Delta x}$ produces the same value. This constant rate of change is precisely what makes a function linear.

Slope-Intercept Form

The most common way to write a linear equation is slope-intercept form:

Here $m$ is the slope and $b$ is the $y$-intercept—the point where the line crosses the $y$-axis, i.e., the value of $y$ when $x = 0$. This form is especially useful for graphing: start at the point $(0, b)$ on the $y$-axis, then use the slope to find additional points by moving $m$ units up (or down if $m$ is negative) for every 1 unit to the right.

Point-Slope Form

When you know the slope $m$ and one point $(x_1, y_1)$ on the line (but the point is not necessarily the $y$-intercept), the most direct way to write the equation is point-slope form:

This form comes directly from the definition of slope. If $(x, y)$ is any other point on the line, then ${\color[rgb]{0.086,0.639,0.290}m} = \frac{y - {\color[rgb]{0.573,0.251,0.055}y_1}}{x - {\color[rgb]{0.573,0.251,0.055}x_1}}$, and multiplying both sides by $(x - {\color[rgb]{0.573,0.251,0.055}x_1})$ gives the point-slope equation. You can always rearrange point-slope form into slope-intercept form by distributing and solving for $y$.

Standard Form

The standard form of a linear equation is:

where $A$, $B$, and $C$ are integers, and by convention $A \geq 0$. Standard form is useful when working with systems of equations (as we will see in Sections 1.3 and 1.4) because it aligns the $x$ and $y$ terms vertically, making elimination straightforward. To convert from slope-intercept form, move the $x$-term to the left side and clear any fractions.

Parallel and Perpendicular Lines

1.2 Graphing Linear Functions

A linear function $f(x) = mx + b$ produces a straight-line graph. While the equation tells you everything algebraically, the graph provides geometric insight that makes many concepts—especially systems of equations—much easier to understand. In this section, we cover three approaches to graphing linear functions and explore piecewise linear functions.

Graphing Using Intercepts

Every non-vertical, non-horizontal line crosses both axes. The two intercepts give us two points, which is all we need to draw the line.

• The $y$-intercept is found by setting $x = 0$: the point is $(0, b)$.
• The $x$-intercept is found by setting $y = 0$ and solving for $x$: the point is $\left(-\frac{b}{m}, 0\right)$ when $m \neq 0$.

Transformations of $y = x$

The parent linear function is $y = x$, which passes through the origin with slope 1. Every other linear function can be obtained from $y = x$ through transformations:

• $y = mx$: a vertical stretch (or compression) of $y = x$ by factor $|m|$, with a reflection across the $x$-axis if $m < 0$. This changes the slope while keeping the line through the origin.
• $y = mx + b$: the same stretch/compression, followed by a vertical shift of $b$ units. This moves the $y$-intercept from $0$ to $b$.

In the interactive graph above, the dashed gray line shows the parent function $y = x$ for reference, while the blue line shows $y = mx + b$. Experiment with different values:

• Set $m = 2$ to see a steeper line; set $m = 0.5$ for a flatter line.
• Set $m = -1$ to see the line reflected across the $x$-axis relative to $y = x$.
• Move $b$ up and down to shift the entire line vertically.
• Set $m = 0$ to see a horizontal line at height $b$.

Piecewise Linear Functions

A piecewise linear function uses different linear formulas on different intervals of the domain. A common example is the absolute value function $f(x) = |x|$, which can be written as:

Piecewise functions arise frequently in real-world modeling. For instance, a cell phone plan might charge \$0.10 per minute for the first 500 minutes and \$0.05 per minute thereafter. To graph a piecewise linear function, graph each piece on its respective interval, using closed dots for included endpoints and open dots for excluded endpoints.

1.3 Systems of Linear Equations (2 Variables)

A system of linear equations consists of two or more linear equations considered simultaneously. The solution of a system is the set of all ordered pairs $(x, y)$ that satisfy every equation in the system at the same time. Geometrically, solving a system of two linear equations means finding the point(s) where two lines intersect.

Figure 1.6 — Three possible outcomes for a 2-variable linear system

The Substitution Method

The idea behind substitution is to solve one equation for one variable and then substitute that expression into the other equation, reducing the system to a single equation in one variable.

The Elimination Method

The elimination method (also called the addition method) works by adding or subtracting the equations to eliminate one variable. If necessary, multiply one or both equations by constants first so that the coefficients of one variable are opposites.

Graphical Interpretation

Graphically, solving a system of two linear equations means finding the intersection point of two lines. The interactive graph below lets you adjust the slopes and intercepts of two lines to see how the intersection changes.

Try these experiments with the graph above:

• Set both slopes equal (e.g., $m_1 = m_2 = 1$) with different intercepts to see parallel lines (no solution).
• Set both slopes equal and both intercepts equal to see the lines overlap (infinitely many solutions).
• Give them different slopes to see a unique intersection point.

1.4 Systems of Linear Equations (3 Variables)

A system of three linear equations in three unknowns has the general form:

Geometrically, each equation represents a plane in three-dimensional space. The solution is the point (or set of points) where all three planes intersect. A unique solution corresponds to the three planes meeting at a single point. If the system is inconsistent, at least two of the planes are parallel (or the three planes form a triangular prism with no common point). If the system is dependent, the planes share a common line or are all the same plane.

Elimination Method for 3 Variables

The strategy is to reduce the 3-variable system to a 2-variable system, then to a single equation. We do this by choosing one variable to eliminate, then pairing equations to create two new equations that do not contain that variable.

Let us now work through a system that does have a unique solution.

1.5 Systems of Linear Inequalities

A linear inequality in two variables looks like $ax + by < c$ (or with $\leq$, $>$, $\geq$). The solution set of a single linear inequality is a half-plane: one side of the corresponding boundary line $ax + by = c$. A system of linear inequalities asks for the set of points that satisfy all the inequalities simultaneously, which is the intersection of the individual half-planes.

Graphing Linear Inequalities

Finding Vertices

The vertices (corner points) of the solution region are found by solving pairs of boundary equations simultaneously. For the example above, the vertices are:

• Intersection of $x + y = 8$ and $2x + y = 6$: subtract to get $x = -2$... but $x \geq 0$, so this intersection is outside the feasible region.
• Intersection of $x + y = 8$ and $x = 0$: gives $(0, 8)$.
• Intersection of $x + y = 8$ and $y = 0$: gives $(8, 0)$.
• Intersection of $2x + y = 6$ and $y = 0$: gives $(3, 0)$.
• Intersection of $2x + y = 6$ and $x = 0$: gives $(0, 6)$.

Checking which vertices satisfy all constraints, the feasible region has vertices at $(3, 0)$, $(8, 0)$, $(0, 8)$, and $(0, 6)$.

Introduction to Linear Programming

1.6 Practice Problems

Test your understanding of the concepts in this chapter. Each problem has a detailed solution that you can reveal after attempting it yourself. Try to work through each problem completely before checking your answer.