Chapter 2.1 — Introduction to Linear Equations
Introduction of linear equations in one variable and their solutions. This is Lesson 1 of 6 in Chapter 2: Linear Equations in One Variable.
What Makes an Equation "Linear"
An equation is a statement that two expressions are equal. Among all equations, the simplest and most useful family is the one where every variable appears only to the first power — no squares, no cubes, no products of two variables multiplied together. An equation like this is called a linear equation, and this chapter is where the systematic tools for solving them are built.
- 2x + 3 = 5 — linear: x appears only to the power 1.
- a + 2b + 9 = 0 — linear: both a and b appear only to the power 1.
- l − 3m = 5n — linear: all three variables appear only to the power 1.
- 5x² + 6xy − 4y² = 16 — not linear: x² and the xy product both push the degree to 2.
- xy + yz + zx = 11 — not linear: every term is a product of two variables.
Narrowing to One Variable
When a linear equation involves only a single variable, it earns a more specific name: a linear equation in one variable, often just called a simple equation. This is the exact form Chapter 2 focuses on. Written in general form, it looks like ax + b = c, where a, b and c are fixed numbers and x is the one unknown quantity being solved for.
ax + b = c (a ≠ 0, one variable x)- 3m + 7 = 13 — a simple equation: only m appears.
- 7y − 4 = 2y + 1 — a simple equation: only y appears, even though it shows up on both sides.
- x/3 − x/5 = 2 — a simple equation: only x appears, even inside fractions.
- 3 = 2x + y — not a simple equation: two variables, x and y, both appear.
- x² + 5x + 3 = 0 — not a simple equation: the x² term makes the degree 2.
- 5m − 6n = 0 — not a simple equation: two variables, m and n, both appear.
A Two-Question Test
Deciding whether a given equation is a linear equation in one variable comes down to two checks, in order: does every variable in the equation appear only to the first power (no squares, cubes, or one variable multiplied by another)? And does exactly one distinct variable appear anywhere in the equation? An equation that passes both checks belongs to this chapter; failing either one — a hidden square, or a second variable — moves it into a different category entirely.
What "Solving" an Equation Actually Means
The solution (or root) of an equation is the specific value of the variable that makes the left-hand side (LHS) exactly equal to the right-hand side (RHS). An equation can look identical for two different candidate values, and only substituting each one in and checking both sides tells you which is correct.
Take 3x − 5 = 7. Try x = 5 first: LHS = 3(5) − 5 = 15 − 5 = 10, while RHS stays 7. Since 10 ≠ 7, x = 5 is not a solution. Now try x = 4: LHS = 3(4) − 5 = 12 − 5 = 7, and RHS = 7. Since LHS now equals RHS, x = 4 is the solution — the one value out of infinitely many possible numbers that balances this particular equation.
Substitute the candidate value → Compute LHS and RHS separately → If they match, it is the solutionWhy Checking Matters More Than It Looks
It's tempting to treat "solving" as purely a mechanical process of moving terms around, with the check at the end as an optional formality. In practice, the check is what catches arithmetic slips made along the way — a sign flipped during transposition, or a term dropped while combining like terms, will usually surface the moment you substitute back in and the two sides fail to match. Building the habit of always substituting the final answer back into the original equation, not a simplified version of it, is worth forming early, since every remaining lesson in this chapter builds directly on it.
A Second Check, With a Twist
Not every candidate value that looks reasonable turns out to be a solution, and a second example makes that concrete. Consider the equation x/2 − 3 = 1. Testing x = 6: LHS = 6/2 − 3 = 3 − 3 = 0, but RHS = 1. Since 0 ≠ 1, x = 6 is not a solution, even though it's a "nice" whole number that might look plausible at a glance. Testing x = 8 instead: LHS = 8/2 − 3 = 4 − 3 = 1, matching RHS exactly, so x = 8 is the solution. Being a round number, or seeming to fit the shape of the equation, is not evidence of anything — only the substitution check settles it.
Why Only One Value Usually Works
A linear equation in one variable, in its general form ax + b = c with a ≠ 0, describes a straight-line relationship between the expression on the left and the fixed number on the right. Because a straight line crosses any single horizontal position exactly once, there is exactly one value of x that balances the equation — not zero, and not several. This is different from equations with higher powers of the variable, which can have more than one solution; part of what makes linear equations approachable is this guarantee of a single, unique answer. It's also why a linear equation is never described as having "multiple solutions" the way some other equations are — finding one correct value is finding the whole answer, not just one of several. That single, guaranteed solution is also what makes the substitution check so reliable — there is never a second "hidden" value lurking that the check might miss — once one substitution confirms LHS = RHS, the search for the solution is genuinely over — there's no need to keep testing further candidate values just in case.
Key Terms to Remember
- Degree of an equation — the highest power any variable is raised to. Linear equations always have degree 1.
- Variable — the unknown quantity, usually written as a letter such as x, y, or m.
- LHS and RHS — the expressions on the left and right of the equals sign.
- Root / Solution — the value of the variable that satisfies the equation, making LHS equal RHS.
- Simple equation — another name for a linear equation in one variable.
The Foundation the Rest of This Chapter Rests On
Recognising what a linear equation in one variable is, and knowing precisely what it means to verify a solution, is the foundation the rest of Chapter 2 is built on. The next step is learning systematic techniques for actually finding that solution — starting with transposing terms — worked through in Exercise 2.1. The same reasoning about variables, degree, and balancing both sides carries forward into Polynomials in Class 9 and Quadratic Equations in Class 10. Since solving equations leans heavily on rational number arithmetic, it's worth revisiting the Introduction to Rational Numbers if any of that feels shaky.