True for some values only Compare these two statements:
\[(x - 2)^2 = x^2 - 4x + 4 \qquad\qquad 3x - 4 = 11\]
The first is true for every value of \(x\) :identity (a special product, in fact). The second is true only when \(x = 5\) :equation in the everyday sense.
DefinitionEquation and solution
An equation states that two expressions are equal. A solution is a value of the variable that makes the statement true. Solving the equation means finding all of its solutions.
The expression on the left of the equals sign is the left side ; the one on the right is the right side . An equation is linear when, after simplifying, the variable appears only to the first power.
The properties of equality Solving means rewriting the equation as simpler and simpler equations with the same solutions, until you reach \(x = \text{a number}\) .
RuleAddition and subtraction property
You can add or subtract the same quantity on both sides.
In practice this means a term can move to the other side if you change its sign :
\[3x - 4 = 11 \quad \Rightarrow \quad 3x = 11 + 4\]
RuleMultiplication and division property
You can multiply or divide both sides by the same nonzero number.
In practice this lets you divide by the variable's coefficient:
\[3x = 15 \quad \Rightarrow \quad x = \frac{15}{3} = 5\]
The same property lets you flip every sign (multiply by \(-1\) )
The method ExampleSolve \(7x + 3 = 4x - 9\)
Move \(4x\) to the left and \(3\) to the right, changing their signs:
\[7x - 4x = -9 - 3\]
\[3x = -12 \quad \Rightarrow \quad x = -4\]
Check: \(7(-4) + 3 = -25\) and \(4(-4) - 9 = -25\) .
ExampleSolve \(5(x - 1) - 2(x + 3) = x + 7\)
Distribute:
\[5x - 5 - 2x - 6 = x + 7\]
Simplify the left side:
\[3x - 11 = x + 7\]
Collect and solve:
\[3x - x = 7 + 11 \quad \Rightarrow \quad 2x = 18 \quad \Rightarrow \quad x = 9\]
ExampleSolve \(\frac{x + 2}{3} - \frac{x}{4} = 2\)
The LCD is \(12\) .
\[4(x + 2) - 3x = 24\]
Keep the parentheses around \(x + 2\) :
\[4x + 8 - 3x = 24 \quad \Rightarrow \quad x = 16\]
Check: \(\frac{18}{3} - \frac{16}{4} = 6 - 4 = 2\) . ✓
ExampleSquares that disappear: \((x - 3)^2 = (x + 1)(x - 1) + 4\)
\[x^2 - 6x + 9 = x^2 - 1 + 4\]
The \(x^2\) terms are the same on both sides, so they cancel:
\[-6x + 9 = 3 \quad \Rightarrow \quad -6x = -6 \quad \Rightarrow \quad x = 1\]
It looked quadratic, but it's linear once simplified.
One solution, no solution, or infinitely many When you reach the form \(ax = b\) ,
Final form What it means Solutions \(a \neq 0\) one solution exactly one: \(x = \frac{b}{a}\) \(0x = b\) with \(b \neq 0\) no solution none \(0x = 0\) identity infinitely many: every real number
ExampleNo solution: \(3(x + 2) = 3x + 1\)
\[3x + 6 = 3x + 1 \quad \Rightarrow \quad 0x = -5\]
No number times zero equals \(-5\) ,
ExampleAn identity: \(2(x - 4) + 3 = 2x - 5\)
\[2x - 8 + 3 = 2x - 5 \quad \Rightarrow \quad 0x = 0\]
Every number works. This equation is an identity in disguise.
Watch outNever divide by zero
From \(0x = -5\) you can't write \(x = \frac{-5}{0}\) :
Practice Quick check
What is the solution of \(6x - 5 = 2x + 11\) ?
A \(x = 2\) B \(x = 4\) C \(x = -4\) D \(x = \frac{3}{2}\)
\(6x - 2x = 11 + 5\) ,\(4x = 16\) and \(x = 4\) .
Quick check
Simplifying an equation, you reach \(0x = 7\) .
A The solution is \(x = 0\) B Every real number is a solution C The equation has no solution D The solution is \(x = 7\)
No number times zero equals 7, so nothing satisfies the equation.