Algebra Equation Guide 2026: Solving Linear and Systems Step-by-Step
Table of Contents
Order of Operations (PEMDAS)
Parentheses, Exponents, Multiplication/Division (left to right), Addition/Subtraction (left to right).
Example: 3 + 4 x 2^2 = 3 + 4 x 4 = 3 + 16 = 19 (not 56).
The #1 algebra error is violating order of operations - when in doubt, add parentheses to force your intended grouping.
The unary minus is PEMDAS's favorite trap: -3^2 = -(3^2) = -9, but (-3)^2 = 9. Exponents bind tighter than the negative sign, so a bare -3^2 squares the 3 and then negates it. Every calculator follows this rule, which is why students who expect 9 get -9 and assume the calculator is broken. Whenever a negative base is raised to a power, parenthesize it - (-3)^2 leaves no room for misreading.
Nested grouping rewards working strictly inside-out. Evaluate 2[3 + 4(5 - 2^2)]: the innermost piece 2^2 = 4 gives 5 - 4 = 1; then 4(1) = 4; then 3 + 4 = 7; finally 2 x 7 = 14. Fraction bars are grouping symbols too: (6 + 4)/(2 + 3) = 10/5 = 2, whereas 6 + 4/2 + 3 without the bar equals 11. If an expression looks ambiguous, add parentheses until it is not.
Solving Linear Equations
Solve: 3x + 7 = 22
- Subtract 7 from both sides: 3x = 15
- Divide by 3: x = 5
- Check: 3(5) + 7 = 22. Correct.
With fractions: (2/3)x - 4 = 8. Add 4: (2/3)x = 12. Multiply by 3/2: x = 18.
Variables both sides: 5x - 3 = 2x + 12. Subtract 2x: 3x - 3 = 12. Add 3: 3x = 15. x = 5.
Literal equations - solving for a letter instead of a number - use every move you already know. Solve P = 2l + 2w for w: subtract 2l from both sides to get P - 2l = 2w, then divide by 2: w = (P - 2l)/2. Nothing changed except the "numbers" are symbols. Physics and finance run on rearranged formulas like this, so treat each letter as a wrapped number and unwrap it with the same inverse operations.
Decimals invite arithmetic slips, so clear them first. For 0.25x + 1.5 = 4.25, multiply every term by 100: 25x + 150 = 425, then 25x = 275 and x = 11. Checking in the original: 0.25(11) + 1.5 = 2.75 + 1.5 = 4.25. Multiplying by the power of 10 that clears the most decimal places turns a fiddly decimal equation into clean integer arithmetic.
Systems of Two Equations
Solve: x + y = 10 and 2x - y = 5
Elimination: Add the equations (y cancels): 3x = 15, x = 5. Substitute: 5 + y = 10, y = 5.
Substitution: From first: y = 10 - x. Into second: 2x - (10 - x) = 5, 3x = 15, x = 5.
Graphically, the solution is where the two lines intersect: point (5, 5).
Elimination exposes degenerate systems instantly. The pair x + y = 5 and 2x + 2y = 12 collapses to 0 = 7 - a contradiction, so the lines are parallel and no solution exists. But x + y = 5 with 2x + 2y = 10 collapses to 0 = 0, meaning the second equation is just the first doubled: infinitely many solutions, one line drawn twice. A false statement means "no solution"; a true statement means "infinitely many."
Ticket word problems are systems in disguise. Adult tickets cost $12, child tickets $7; 200 tickets sold for $1,850. Let a = adults, c = children: a + c = 200 and 12a + 7c = 1850. Substituting c = 200 - a gives 12a + 7(200 - a) = 1850, so 5a + 1400 = 1850, 5a = 450, a = 90, c = 110. Verify: 90 x 12 + 110 x 7 = 1080 + 770 = 1850. Two unknowns, two facts, one substitution.
Real-World Applications
- Break-even: Cost = 5000 + 20x, Revenue = 50x. Set equal: 5000 + 20x = 50x, 30x = 5000, x = 167 units.
- Mixture: How much 20% solution + 50% solution makes 10L of 30%? x + y = 10, 0.2x + 0.5y = 3. Solve: 6.67L of 20%, 3.33L of 50%.
- Rate problems: Two trains, speeds, meeting times - all systems of equations.
Straight-line depreciation is a linear equation wearing an accounting suit. A $24,000 car losing $2,500 per year has value V = 24000 - 2500t after t years. Setting V = 9000: 24000 - 2500t = 9000, so 2500t = 15000 and t = 6 years. Businesses use exactly this model to schedule equipment replacement - the "when is it worth less than the repair cost" question is always a linear equation.
Every "which plan is cheaper" question is a system waiting to happen. Plan A charges $40 + $5/GB; Plan B charges $55 + $2/GB. Set them equal: 40 + 5x = 55 + 2x, giving 3x = 15 and x = 5 GB. Below 5 GB choose A; above it choose B. The intersection point of the two cost lines is the entire decision - one equation replaces a whole spreadsheet of what-ifs.
Frequently Asked Questions
Whatever you do to one side, do to the other. Every step must preserve the equality.
Multiply everything by the common denominator first to clear fractions, then solve normally.
Elimination when coefficients align nicely (like +y and -y); substitution when one variable is already isolated. Both always work.
Parallel lines (same slope, different intercept) never meet - e.g., x + y = 5 and x + y = 7. Algebraically you get a contradiction like 0 = 2.
The equations describe the same line (one is a multiple of the other) - every point on the line is a solution.