Key takeaway: A linear equation is an equation whose graph is a straight line, with variables that appear only to the first power and are never multiplied together.
Grade 88.EE.C.7Expressions and Equations
Definition of Linear Equation
A linear equation describes a straight-line relationship. Every ordered pair that makes it true is a point on the line, and every point of the line makes it true, so the graph is simply a picture of all the solutions.
With one variable, such as \(4x – 7 = 21\), there is a single solution. With two variables, such as \(3x – 2y = 12\), there are infinitely many solutions, one for each point of the line. The key test is the form: each variable has an exponent of 1 and appears alone, not inside a root, a denominator or a product with another variable. Equations that break these rules graph as curves.
Formula and notation
Two common ways to write a two-variable version are
\[Ax + By = C \qquad \text{and} \qquad y = mx + b\]
with \(A\) and \(B\) not both zero. Substitute an ordered pair \((p, q)\) to test whether it is a solution.
Key facts
- The graph is a straight line, whether it rises, falls or lies flat.
- Variables have exponent 1 and are never multiplied by each other.
- Plugging in x = 0 and y = 0 gives the two axis crossings quickly.
- Two points are enough to draw the whole line.
Linear Equation example
Which of the points \((5, 1)\) and \((8, 6)\) is a solution of \(3x – 2y = 12\)? Then find the intercepts.
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Test \((5, 1)\): \(3(5) – 2(1) = 15 – 2 = 13\), which is not 12.
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Test \((8, 6)\): \(3(8) – 2(6) = 24 – 12 = 12\). It works.
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x-intercept: set \(y = 0\): \(3x = 12\), so \(x = 4\). y-intercept: set \(x = 0\): \(-2y = 12\), so \(y = -6\).
Only \((8, 6)\) is a solution, and the line crosses the axes at \((4, 0)\) and \((0, -6)\).
Common mistake
Watch out
Mateo says that \(y = x^{2} + 1\) is a linear equation because it has an x and a y and a plus sign.
Why it is wrong
The exponent on \(x\) is 2, so the graph is a U-shaped curve, not a line. Check with three points: \((0, 1)\), \((1, 2)\) and \((2, 5)\) have unequal steps in y for equal steps in x. A line must change by the same amount each time.
Related terms and lessons
- Slope-Intercept Form
- One-Step Equation
- System of Equations
- Linear Function
- Standard Form of a Linear Equation
- Grade 8 · Solving Linear EquationsLessonWorksheetQuiz
- Algebra 1 · Linear Equations and Literal EquationsLessonWorksheetQuiz
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