In This Lesson What Is an Equation? Linear Equations Quadratic Equations Systems of Equations Absolute Value Equations Radical Equations What Is an Equation? An equation is a mathematical statement that two expressions are equal, connected by the "=" sign. Solving an equation means finding all values of the variable(s) that make the statement true.
3x + 5 = 20
The fundamental principle of equation solving: whatever you do to one side, you must do to the other . This preserves the equality while isolating the unknown.
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Solving One-Variable Linear Equations Example: Solve 4x − 7 = 13 Step 1: Add 7 to both sides: 4x = 20
Step 2: Divide both sides by 4: x = 5
Check: 4(5) − 7 = 20 − 7 = 13 ✓
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Step 2: Subtract 3: 3x = 15
Step 3: Divide by 3: x = 5
Equations with Fractions xBZqdZD2quD3xeN271icbHcNOCI0CJALuWg66o1Z5lD2NLegSlw7piBki1tfnkDtgTu4CnU+wLvM1Tu1J8QBq+dcnC3w6c++lZBv6mLmSkoKZ503CdF0gAuntHnn8zD8QoJWy2wjRR4NFBH0/HGXKEIiQSVPiP3ZVMIZdQMBQY8WZfEmRcnsj6qYxxQWrI32OYo3UH8q4jtcCntGMIzh3hf5Qfmg53/bLBOUmCCQ5p2+dV03Rpi1oPnmdKKO17foZwwbBvs4/jTReJ+XrIbaUgYtw8sKYzBYdt+S1MYpmzYiRcoR6p08dbnEH5JAhLFcMPdPlRIpdOMtYewW8DbtATtnzMGo4SuPjHnpQ1I1auNn/x/BeDjyMQ3z5+DPN5ERdtY3oeWnA3E/oEPYbsDdSMjmKSn/bAUhb0KK2KQInOyIJq+rI8/bqQGEvaSdk1JPoiH1zWaWSsCsoAYBEUHni9IG2+Pg86QLL/ss2g6CLpZyel8xGZySnWzfYf8/x/X2VvSmx5lllbVfZ1q4vuOrxpDf4LNdOF2HjK9YftxkXuxuU8bm9iL/mNXSuZNmmBSvyZnP9XJqthmed8ll2fdqqJq6ghKYfPOMjg2bNWHo8ScGj9L89HacP+RsNbZEdvF1D9HI1bizkqZEgC1FBTxogtuCYuEl5fdCihBHLREg6YNFseqwHM0UjbBBfmdPD5EmitT1thermkz9voCNkKmu4U0fCICwZyLeuyV7ad4S3a5tl5bIOLuNOD00NwVpOrYby8a+o7e76WtZoENtPaUenxuoNkBRZfswAdA7F When an equation contains fractions, multiply every term by the least common denominator (LCD) to clear the fractions first.
Example: Solve x/3 + x/4 = 7 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 LCD = 12. Multiply every term by 12:
4x + 3x = 84
7x = 84 → x = 12
Linear equations connect directly to linear functions , whose graphs are straight lines with slope m and y-intercept b in the form y = mx + b.
Quadratic Equations jMm29YSKx7cJdIK5W8nf0v/wVBpPAyd9Q2N4kuQ5mEVelghz+FREG/09LFq7si2Yw8bhHOWKDAdBn712T/V1v/gQdWyIDMjq0atvZUQJ5eGVNJeGwTr90nMGgVSLtIj/mcfXC8LZvNtryoi8UmXHx12VPR1w+SLbJujI5AF8CkixnFqAIWifUdfPrc7Z/KQGtypLiaja5j0HQABpcAlhEcoQh9RZWS0ExE4EoXY6yaYTvAYHsK1jYTOCimp4CYDDgB29+/nwm6QbtAOMFzSPHgp97V4bUSW1k9PIdxVlWGladxooGBTiulbp83fOMHKxjAgMtQtzf3WHTXSZ1B019bBLVP2oh4MxSMh4SHd8qoIo5cdql8S5P7qVfQ8GNmd7cczqdEckpy06xLmnRG4UXUPLXgh0zfyQ0gV97Y9uVZorK3R/+SsLlBnxSHKNqOj/HBczzdHVH1OKZJr8CDuZWxPRAvYFO3tJuY4jjNNbl1YITmb3n9ZBLW9j7VcDw7CtByyqNAx8BKBC5Dd9xIjtWBCbJOE7vLlLJB/2mnwLnsNfihAOwyqAD7tY+pCukDjyK66G8UMcE+Th+plZ87mSIAjY6vWbPoQl6mDfUOEFMVygcuR/brcOmdN5bZGfEtVtnvPB+Xd4NWbnGSxSmYzDFoxH9afmmTZZmvRTJ3E5UNTOAouBQLZwZPZrssqmfib+V8ovMNA633f19XaJLWlBpYT+RYSDrRBHGm/24PeXfsln1d+yVgwhhCLVmU9N9lu/ZTd56fAR7TCL437DPMuJmC2qpbPkhEfiIxya A quadratic equation has the standard form ax² + bx + c = 0 (where a ≠ 0). These equations can have 0, 1, or 2 real solutions.
Method 1: Factoring If you can factor the quadratic, set each factor equal to zero (see the Polynomials & Factoring page for more techniques).
Example: Solve x² − 5x + 6 = 0 Factor: (x − 2)(x − 3) = 0
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Method 2: The Quadratic Formula
x = (−b ± √(b² − 4ac)) / (2a)
The expression Δ = b² − 4ac is called the discriminant . It tells you the nature of the solutions:
Δ > 0: Two distinct real solutionsΔ = 0: One repeated real solutionΔ < 0: Two complex conjugate solutions (see complex numbers ) Example: Solve 2x² + 3x − 5 = 0 a = 2, b = 3, c = −5
Δ = 9 − 4(2)(−5) = 9 + 40 = 49
x = (−3 ± 7) / 4
x = 1 or x = −5/2
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Example: Solve x² + 6x + 2 = 0 2lcJTQf4O71FMZaEs8IVThqdVWbCTUWvgMY9LRqPviRtqRBWh50e2iKBH3gU54uQ3JGip7qvLv5KqGSyTbOQFJdpUdjKptvHokxJ4KFOrZgPZdt4UWvlxl2D9VJPaFwJEZNa1aEjQHhM+1Xh/EaRwbfSXecRnBqfXnJEQyMn3Z4iiBcrfJIXcZZpVoOLLhaoc57gPVtPUJTPh8+AqF4HZDVrtlGgDphN8MivP9iwv5D+H7a+hpyQsFKLcsShBGr/HJRp3+EXyOy0zKNGQ2wEI7xYN4eeiCqtLSD788E+5nrG1MIC3blFOjEYC6HTdY1ichb3lAmz1O2WZjl02xS8hHTrhuubRdzlNK5SsvXFAJVH9f2/go7O3kXhCK1dlnxndaD+yvBbPk204a9nXPYeEzfYVz3R4q3ZoyisOm0vS3SYyDZVwOoEyAkt1i6kSiumdJzIs8KrvrIJWHv+hTwmSfR6sguyoQLs4i83gVDFnBKI4Q3588RkzOINkr3OyJPyipU/ddoLHsk7F4/RLlqtOmbDfkQXee9I180kfntbrGIW4HtLATsw/8xrKHl/lqpIwpOS+ENdAEeI9pKh7NU8+rjD6UbgLc6+CFEuJny/B1NRt/efVvumWyk1JVUlx+ZTOaOJU4Dyxj40NulIWxPbLA2GDpgkrVQ4tbT7UB/FPSwx6Zq/sO7IzP7Acnhy4T2UpN8xEHmMrOBGUzB/lcz5meh/985S74YaJQSP2e5irJAziB4nmFfZLggMGyd8Q2J40u856BxOaw1N5lQ+xMEVLLo10SqtfkSJJmub x² + 6x = −2
x² + 6x + 9 = −2 + 9 (add (6/2)² = 9 to both sides)
(x + 3)² = 7
x = −3 ± √7
Systems of Equations A system of equations is a set of two or more equations with the same variables. The solution is the set of values that satisfies all equations simultaneously.
Method 1: Substitution Example: Solve the system y = 2x + 1
3x + y = 11
Substitute: 3x + (2x + 1) = 11 → 5x = 10 → x = 2, y = 5
Method 2: Elimination Example: Solve the system 2x + 3y = 12
4x − 3y = 6
Add the equations: 6x = 18 → x = 3, then y = 2
Method 3: Matrices (Cramer's Rule) For larger systems, matrix methods are far more efficient. See Linear Algebra for Gaussian elimination, and the formula sheet for Cramer's Rule.
Systems of equations connect algebra to
linear algebra , where Ax = b is the central problem. In higher dimensions, you can't solve systems by hand — you need
matrix methods .
Absolute Value Equations The absolute value |x| gives the distance of x from zero. To solve |expression| = k (where k ≥ 0), split into two cases:
|ax + b| = k ⟹ ax + b = k or ax + b = −k
Example: Solve |2x − 5| = 9 Case 1: 2x − 5 = 9 → x = 7
Case 2: 2x − 5 = −9 → x = −2
Solution: x = 7 or x = −2
Radical Equations A radical equation contains a variable inside a radical (√). Isolate the radical and square both sides — but always check for extraneous solutions!
Example: Solve √(x + 3) = x − 1 Square both sides: x + 3 = x² − 2x + 1
Rearrange: x² − 3x − 2 = 0
Factor/quadratic formula: x = (3 ± √17)/2
Check both in the original equation — reject any that produce a false statement.
Radical equations will reappear in
trigonometry (half-angle formulas) and
integration (u-substitution with radicals).