Linear Equations in One Variable: Standard Form, Solving Steps, and Examples

She works $h$ hours at 15 dollars an hour, on top of 400 dollars banked; the trip costs 2,500 dollars. The whole plan is one line: $400+15h=2500$.

The scale below draws that line: left pan, the two money terms; right pan, the trip price. Linear equations in one variable are exactly this: two sides that one value of one letter can level. Here that value is $h=140$, since $400+15\times140=2500$ - and every budget of this shape runs on linear equations in one variable.

A linear equation in one variable is an equation that can be written as $ax+b=0$, where $a$ and $b$ are real numbers and $a\neq0$; the only power of the variable is 1. Ray's 1866 New Higher Algebra agrees: a simple equation of the first degree contains no power of the unknown higher than the first. That is the membership test for linear equations in one variable.

Test any candidate against three features.

1. One variable only. $3x+7=22$ carries the single letter $x$; $2x+5y=1$ carries two and belongs elsewhere.
2. Degree 1. The unknown is never squared or cubed. Compare $x^2=9$: the power 2 makes it a quadratic.
3. Standard form $ax+b=0$. Every member rearranges to it. From $3x+7=22$: $3x-15=0$, so $a=3$, $b=-15$.

Ray's roles: knowns take numbers or first letters $a$, $b$, $c$; the unknown takes a last letter such as $x$. Linear equations in one variable keep exactly one such unknown on stage.

Fail cases: $\frac{3}{x}=6$ hides the unknown in a denominator; $x^2+1=10$ strikes out on degree. One strike, and the equation leaves the family of linear equations in one variable. That strike rule is what keeps linear equations in one variable a tight family.

Membership in the family of linear equations in one variable, sorted at a glance:

Linear equations in one variable pass every row's test: rearrange to $ax+b=0$ - one unknown, highest power 1. A system of two linear equations lives on another page.

The goal: isolate the variable until one side holds only it and the other only a number. Ray's rule book pairs each connection with its undo:

The same moves in order - simplify each side ($a(b+c)=ab+ac$), gather variable terms, undo multiplication, never divide by zero.

Ray closes with a habit: substitute the answer back; if both members match, it is true. That is the method for linear equations in one variable - undo, then verify.

"Solve for" sounds like a separate skill, yet with linear equations in one variable it is the same balance work. The four properties of equality - add, subtract, multiply, or divide the same amount on both sides - each preserve the claim. Linear equations in one variable never ask for more.

One caution: operate on every term of a side. Clearing fractions in $2+x=\frac{x}{3}$ gives $3(2+x)=x$, not $6+x=x$ - parentheses first.

Fractions invite one more choice: coefficients only - wait to the last step; fractions everywhere - clear them at once. Pick one road.

Two moves, one drill. Take $5x-3=2x+9$: subtract $2x$, get $3x-3=9$; subtract 9, get $3x-12=0$ - so $a=3$, $b=-12$. Take $4-x=2x+10$: gather terms, get $-3x-6=0$, so $a=-3$, $b=-6$. Both are linear equations in one variable wearing everyday clothes. The drill is the reason linear equations in one variable feel alike: every one folds into $ax+b=0$, and linear equations in one variable never fold any other way.

Problem. Solve $3x+7=22$, then $2x+1=-9$.

Step 1. Subtract 7 from both sides: $3x=15$.

Step 2. Divide by 3: $x=5$.

Check (Ray's verification). $3\times5+7=22$ - the members match.

For $2x+1=-9$: subtract 1, so $2x=-10$; divide by 2, so $x=-5$. Linear equations in one variable rarely get simpler than this - one unknown, degree 1, two undo moves. Once checked, linear equations in one variable are safe to trust - two solves, two checks, both ends covered. The habit is cheap; linear equations in one variable repay it every time.

Problem. Solve $3(x+4)-5=2x+5$.

Step 1. Distribute: $3x+12-5=2x+5$, so $3x+7=2x+5$.

Step 2. Subtract $2x$ from both sides: $x+7=5$.

Step 3. Subtract 7: $x=-2$.

Check. Left: $3(-2+4)-5=1$; right: $2(-2)+5=1$. Both equal 1, so $x=-2$ stands - linear equations in one variable moved both sides together, and the balance held. With variables on both sides, linear equations in one variable reward one habit: gather the $x$ terms first. The habit scales - every harder linear equations in one variable chapter, fractions included, builds on it...

Problem. Sort three linear equations in one variable: $5x+2=3x+8$, then $4x-10=4(x-3)$, then $5x=2x+3x$.

Case 1. $5x+2=3x+8$: subtract $3x$, get $2x+2=8$, so $x=3$ - a conditional, true for only some values.

Case 2. $4x-10=4(x-3)$: distribute to $4x-10=4x-12$; subtract $4x$, and $-10\neq-12$ survives as false claim. Inconsistent - no solution.

Case 3. $5x=2x+3x$: subtract $2x$ and $3x$, reach $0=0$ - true no matter what. An identity: every real number works.

So linear equations in one variable end three ways: one value, no value, or all values. Across the family, the sort works the same for every linear equations in one variable candidate: simplify, compare, read the verdict.

The two special endings of linear equations in one variable, side by side. $4x-10=4(x-3)$ collapses to $-10\neq-12$: false claim, no solution. $5x=2x+3x$ collapses to $0=0$: always true, every number works. For linear equations in one variable, what survives after simplifying tells all - the survivors are the verdict, and Three labels cover every linear equations in one variable outcome.

Take $2x+1=-9$ with answer $x=-5$: substitute back, $2(-5)+1=-9$, members match - done. Linear equations in one variable come with this built-in referee: every solve of linear equations in one variable ends as substitute, compare, declare. Once checked, linear equations in one variable are safe to carry into homework.

Linear equations in one variable forgive almost nothing - grade yourself on the classic five slips.

1. One-sided operations. Subtract 7 from the left of $3x+7=22$ only, and the balance dies. Fix: both sides or neither.

2. Dropping a term when clearing fractions. In $2+x=\frac{x}{3}$, multiplying by 3 must give $3(2+x)=x$. Fix: wrap the side in parentheses first.

3. Flubbing signs with negatives. $-2(x-5)=-2x+10$, not $-2x-10$. Fix: distribute the sign too.

4. Dividing by zero or by the unknown. From $ax=b$, divide by $a$ only when $a\neq0$ - never by the unknown itself. Fix: divide by known numbers only.

5. Skipping the check. Ray called this verification. Fix: substitute the value back; both members must match. The check keeps linear equations in one variable honest - keep it, and linear equations in one variable stop being a guessing game.