Solving Equations
1.4 Solving Equations
What Solving Means
An equation is a balance statement. Solving is finding every value that keeps that balance true.
Definition: Equation and Solution
An equation states that two expressions are equal. A solution is a value of the variable that makes the equation true.
Fact: Balance principle
If two expressions are equal, adding, subtracting, multiplying, or dividing both sides by the same nonzero quantity preserves equality.
Example: Solve a one-step equation
Solve:
$$x-9=14.$$
Show Solution
Add 9 to both sides:
$$x-9+9=14+9\Rightarrow x=23.$$
Check: $23-9=14$, true.
MyOpenMath: Practice: one-step and two-step equations (placeholder)
Linear Equations
Linear equations often take multiple cleanup steps. The strategy is still the same: simplify each side, then isolate the variable.
Definition: Linear Equation
A linear equation in one variable can be written in the form
$$ax+b=0,\qquad a\ne 0.$$
Example: Solve with distribution
Solve:
$$3(2x-5)-4=2x+7.$$
Show Solution
Distribute on the left:
$$6x-15-4=2x+7\Rightarrow 6x-19=2x+7.$$
Move variable terms to one side:
$$4x-19=7.$$
Add 19:
$$4x=26.$$
Divide by 4:
$$x=\frac{26}{4}=\frac{13}{2}. $$
Final answer:
$$x=\frac{13}{2}. $$
Example: Solve with fractions
Solve:
$$\frac{x}{3}+\frac{5}{6}=2.$$
Show Solution
Subtract $\frac{5}{6}$:
$$\frac{x}{3}=2-\frac{5}{6}=\frac{12}{6}-\frac{5}{6}=\frac{7}{6}. $$
Multiply both sides by 3:
$$x=\frac{7}{6}\cdot 3=\frac{7}{2}. $$
Final answer:
$$x=\frac{7}{2}. $$
MyOpenMath: Practice: linear equations with distribution and fractions (placeholder)
Rational Equations
Rational equations include fractions with variables. The fastest method is to clear denominators using the least common denominator (LCD).
Definition: Rational Equation
A rational equation contains one or more rational expressions with variables in denominators.
Fact: LCD method
For a rational equation:
- Find the LCD of all denominators.
- Multiply every term by the LCD.
- Solve the resulting equation.
- Check for excluded values and extraneous solutions.
Example: Solve a rational equation
Solve:
$$\frac{7}{2x}-\frac{5}{3x}=\frac{22}{3}. $$
Show Solution
Denominators are $2x$, $3x$, and $3$, so an LCD is $6x$.
Multiply each term by $6x$:
$$6x\cdot\frac{7}{2x}-6x\cdot\frac{5}{3x}=6x\cdot\frac{22}{3}. $$
Simplify:
$$21-10=44x\Rightarrow 11=44x\Rightarrow x=\frac{1}{4}. $$
Check denominator restriction: $x\ne 0$, and $\frac14$ is allowed.
Final answer:
$$x=\frac{1}{4}. $$
