Linear Equations Express Balance

An equation states that two expressions have the same value. Solving means finding every value that makes that statement true. The central principle is balance: perform an operation that preserves the solution set.

Learning objectives

You will solve and classify linear equations, handle fractions, verify solutions, and translate a word relationship into an equation.

Why doing the same thing works

If $A=B$, then $A+C=B+C$. Multiplying both sides by the same nonzero number also preserves equality. These properties—not a rule about “moving terms”—justify equation solving.

Fractions: clear denominators legally

For

$$\frac{x-1}{3}+\frac{x+2}{2}=5,$$

multiply every term by the least common denominator $6$:

$2(x-1)+3(x+2)=30.$

Then $5x+4=30$ and $x=26/5$.

One, none, or infinitely many solutions

Simplifying $2(x+1)=2x+2$ produces $2=2$: every real $x$ works. Simplifying $2(x+1)=2x+3$ produces $2=3$: no value works. These outcomes describe the original equations; they are not failed calculations.

Model a situation

A taxi charges a fixed $4$ dollars plus $2.50$ dollars per mile. A $19$-dollar fare satisfies $4+2.5m=19$, so $m=6$. Units validate the structure: dollars plus dollars per mile times miles gives dollars.

Check your understanding

Solve $5-2(3x+1)=4x-7$ and verify.

Show the reasoning

$5-6x-2=4x-7$, so $3-6x=4x-7$, $10=10x$, and $x=1$. Substitution gives $-3$ on both sides.