The Straight Path of Math: Demystifying Linear Equations
What Makes an Equation "Linear"?
The core rule is simple: the variable (or variables) in a linear equation must have an exponent of exactly one. This means you will see $x$, but never $x^2$, $\sqrt{x}$ (which is $x^{1/2}$), or $\frac{1}{x}$ (which is $x^{-1}$). The variable is not multiplied by itself, nor is it inside a function like sine or logarithm. This restriction leads to the standard forms of a linear equation.
• One Variable: $ax + b = 0$
• Two Variables (Standard Form): $Ax + By = C$
• Two Variables (Slope-Intercept Form): $y = mx + b$
Here, $x$ and $y$ are variables, and $a, b, A, B, C, m$ are constants (numbers).
Let's look at some clear examples and non-examples to solidify this concept.
| Equation | Linear? | Reason |
|---|---|---|
| $3x + 5 = 11$ | Yes | Variable $x$ has power 1. |
| $y = -\frac{2}{3}x + 4$ | Yes | It's in slope-intercept form. $x$ is to the first power. |
| $2x - 7y = 15$ | Yes | Both $x$ and $y$ have power 1 (Standard Form). |
| $x^2 + 3x - 4 = 0$ | No | Contains $x^2$ (power of 2). This is a quadratic equation. |
| $5xy = 10$ | No | Variables $x$ and $y$ are multiplied together. The term $xy$ has a combined degree of 2. |
| $\frac{5}{x} = 2$ | No | The variable is in the denominator, equivalent to $5x^{-1} = 2$, which has a power of -1. |
Solving the One-Variable Case
Solving a linear equation in one variable, like $3x + 5 = 11$, means finding the value of $x$ that makes the equation true. This value is called the solution or root. The process involves using inverse operations to isolate the variable on one side of the equals sign. The key is to perform the same operation to both sides to maintain balance.
Example: Solve $2x - 7 = 15$.
Step 1: Isolate the term with the variable. Add 7 to both sides to undo subtraction.
$2x - 7 + 7 = 15 + 7$
$2x = 22$
Step 2: Isolate the variable itself. Divide both sides by 2 to undo multiplication.
$\frac{2x}{2} = \frac{22}{2}$
$x = 11$
Step 3: Check the solution by plugging $x = 11$ back into the original equation.
$2(11) - 7 = 22 - 7 = 15$. This is correct, so $x = 11$ is the solution.
The Graph: Why "Linear" Means "Line"
The most powerful visual representation of a two-variable linear equation is its graph. Because the variable $x$ is only to the first power, the relationship between $x$ and $y$ changes at a constant rate. Plotting all the $(x, y)$ pairs that satisfy the equation always results in a perfectly straight line.
Consider the equation $y = 2x + 1$. This is in the slope-intercept form $y = mx + b$.
- Slope ($m$): The number multiplying $x$. Here, $m=2$. It tells us the line rises 2 units for every 1 unit it runs to the right. It represents the rate of change.
- Y-intercept ($b$): The constant term. Here, $b=1$. It is the point where the line crosses the y-axis $(0,1)$.
If the exponent on $x$ were 2, the rate of change would not be constant. The graph would curve (a parabola), which is why it's called a non-linear equation.
Linear Equations in the Real World
Linear equations are not just abstract math problems; they model countless real-life situations with a constant rate. Here are two concrete examples.
Example 1: Phone Bill Plan. Imagine a phone plan charges a flat monthly fee of $20$ plus $0.10$ per minute of call time. The total monthly cost $C$ depends on the minutes used $m$. This is a linear relationship:
$C = 0.10m + 20$.
The slope is $0.10$ (cost per minute), and the y-intercept is $20$ (the fixed fee). If you use 100 minutes, your cost is $C = 0.10(100) + 20 = 10 + 20 = 30$ dollars.
Example 2: Saving Money. You have $50$ in your savings and decide to save $15$ each week. The total savings $S$ after $w$ weeks is:
$S = 15w + 50$.
This linear equation helps predict future savings. After 10 weeks: $S = 15(10) + 50 = 200$ dollars.
Important Questions
Yes. A linear equation can have two, three, or even more variables. The rule still applies: each variable must be to the first power and not multiplied by another variable. Examples include $2x + 3y = 7$ (two variables) and $x - y + 5z = 0$ (three variables). The graph of a two-variable linear equation is a line in a plane, while a three-variable equation graphs as a plane in three-dimensional space.
Absolutely. This is a linear equation in one variable. It can be thought of as $1 \cdot x + 0 = 5$, which fits the form $ax + b = 0$ (after subtracting 5 from both sides: $x - 5 = 0$). Its graph on a number line is a single point at 5. On an xy-coordinate plane, the equation $x = 5$ graphs as a vertical line where the x-coordinate is always 5.
If the slope is zero, the equation becomes $y = b$. This is still a linear equation. It describes a horizontal line where the y-value is constant no matter what $x$ is. For example, $y = -3$ is a horizontal line crossing the y-axis at -3. It represents a situation with no change, like a fixed flat fee with no per-unit cost.
Footnote
1 Slope: A number that describes the steepness and direction of a line, calculated as the ratio of the vertical change (rise) to the horizontal change (run) between any two points on the line.
2 Y-intercept: The point where a graph intersects (crosses) the y-axis. Its coordinates are always $(0, b)$.
3 Quadratic Equation: An equation where the highest power of the variable is two (e.g., $ax^2 + bx + c = 0$). Its graph is a curve called a parabola.
4 Root (of an equation): A value that, when substituted for the variable, makes the equation true. Also called a solution.
5 Rate of Change: How much one quantity changes, on average, relative to the change in another quantity. In a linear relationship, the rate of change is constant and equal to the slope.