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.
Linear Equations A linear equation in one variable has the form ax + b = c , where the variable x appears only to the first power. The graph of a linear equation in two variables is always a straight line (hence the name).
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 ✓
Equations with Variables on Both Sides Example: Solve 5x + 3 = 2x + 18 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 Step 1: Subtract 2x from both sides: 3x + 3 = 18
Step 2: Subtract 3: 3x = 15
Step 3: Divide by 3: x = 5
Equations with Fractions ZGMurbjMvi+mpN67fII1/G5KBY6PVbm1dtLDa7gzEdQS8B6RW4Kckkfli03SvviCw4AW/lXZUsm+ve2WfCiWlNM/zLNsHdN8YBr1POtCBvMK0GooG7hQPEZyJvZDIYUrgdLxaQJyfmMiKr8YZte4S8sDO5kbTVaOLASBSlAc1Roxw9OzG8DJK1Bs3T6p4po9vi6Sxt8ag81yKDLWXtwRXaJU406tgcMTUYmqzzsNVMEzYVzLWGgd5FOREJ8BdVEzyhh069we0Zyrg8G4c0erU9mcy8vS8Se3FXxOA61+ewVIPSZgZ5nHwblcEBRPDi8JxhM2g0VAnUlz6InnPc//QVm7Ee2OrjMXs7AoLjMAZVRZw1p3bq6MhnZ9wo4bCz/k+FFEv3lPmvyfqbZoHrv17kxNqlIRVRexLBXjPxmfQxDfK+WsSPGYUXkiZI3xT6OU4i4860gxFqbyNSY1fd+JA0rPFO28t268cT2HCCcSobv9l6IlVhLpdaKAp7cgjvOotps7MVYd6r8yz+pV5IhW+u8ZQV2eY+gGhr9izXrLq4x7QhrT92HutvXD7AfdN6h5scD9C8yCS9S1rUYFUZ+7ZvVslEavON14Qq/QCifkO8Q3ZjY0ODib5+oVBn/4PudhKJfBVTNdzkzBAHtI+FDGalv1XxWfRcPnPV5yqDqAMGb3De4HM0HknwU0wxD8KBc7y6Eg2yncJ9wO+ycwygrc8jcyisg5H0QVa2FuhMc3b5VyTvkU9qMHbRse58dWlkDJyMpybkME7gBLr9JccFBG+VCglp8ssEsUbq9 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 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 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
x − 2 = 0 → x = 2 or x − 3 = 0 → x = 3
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 solution0eBLecx1S22hagb9Cx1QIE+q1eD4BnEwBA04gU3yyZPZYXdg1EIpKtYPQkN+T3rPk+TbYKXVAUS6P6XEHcJS2K+fQ7PMnYinYhpqjADvVvrZrRPo3qjiNfY+KPto9A5mCJ4f/aqj/Bjq2vL/OOb88FY+45ARa66kEn6qS6kRRaY3XH1GIOucHOfg65hsOZISiP5HlU8+CujAq9nqsn9JDnzDTldPFiQs0QNN4/8SiGgJysAM6TDz1WR5Qm9Zr+pKZeXhu5jtMix2E3dJIMbVkf5JaPzmAWE1vsMIDumu10IN6RJkvC448Vr6LXwVCiWOxrapMxRTx5rqHy2D9waELICi/1a+2agvPyKEj9Pxciyd4Ul/54LxC9Y+A493qV+LMO9dXW9sftwQ9HO1/NVJVLcys/Wnv0qxSCKp2thugeZamZt4jNIegDg8227TWtuQRp2qEHmkUPv4c0QJlh15F0+jJBdbNfbXzEL0h72igQeRBkl4i/qsQRg4QQsrcNWx1Oo2zJvEu0da51WXnRCCRAZIm4LOJ03opX9WkZaJ134JHxToPJ4ZU2jRlURsw7i0aoRCP9Ch8x6ZSMsSsLNV3UFlZKNV1NthFYn3pDBww+JuhtfArQbPEHo+KUpBmDe7paiMSkVXMyhGtbkX1XcGivl7JG8LhX2HUmSnAzl0fMiwWD5AJBjWutqrCe7QrA+mhT/88nQG+gINjtd1nPrnfktk8Bq5ITPTh/TU7x2F9+kr/lI038Q1KokUObMYMNqIDQO8JJMVxMkxaA1okdbs/bnv/ARpeHDac3a Δ < 0: Two complex conjugate solutions (see complex numbers )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 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
Method 3: Completing the Square This technique rewrites ax² + bx + c as a(x − h)² + k, revealing the vertex of the parabola .
Example: Solve x² + 6x + 2 = 0 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 x0c5J9YUy3XJVxd/DLdwsBGdbceOKMBvGFDw7tYYM++VIQ2JXt7A95GhQVWWlN/faJnXNYAX7uh9YnqAJDBC+Y8GtBvB42DcC4/O0IEhwvEmhZcodPAhk7LNCKtGYPHeHMjkKVcTosnrjGH65YMQE1OXU8ush4LPRmyHmnJCPOV3Ez7ri+8DsYN113TrxsrF1jJ4TXafLUHIfDKJtmfnLMbi6y/i0plPhdv7qi1YwAZSH4v4PGy65HgFpwfmScgPt0WWiF/o941gAh6bdeINsejZhQglIASViQA5Djyui67MQ3TidiHY7xQv6nOBltNlZZpRMAouBbKChzh0sQCJ9bzoE01P0yKiTkorlIiP8iRBGLAT3iH8xagXkcrutGBv8Enr3czdhrOxJcX4iWf/dOOF9PWReRjqH1d4ziok8Bh3ww1ZwoJBakdLw7RziIJTocex5HgpK49cbddzK9D9ISsjmNabbxXt5gFDqukRALXBnUV3EtAthu1jkBI6dw/NoQFCP/yeEstgSmPhzSyPuNcGrbF4TkdV0DHNgydJMO+Nr4Z1OIMBokbbREZShtAV3FfYSkXWX5JhT1GHbPDha97L1afw+CMVZPez4My11fZp/WfEW8F8vNSccrHqkS82+gduke6yKCF9KOdFvtGB0/e4l2ibQJuIAyTu3kZYnA0EaQ1oJSifzbalRsjKTrfzw9QJ/b1W4Xds/AUBkfUrgMyOJbZyK/oXPBQYWH6aVn2HR6zKifAW3EqyYWbjHgwmwfUoGilus2rc2KZmU4SZXwwOXHgMwRdQAQt 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 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 Example: Solve the system 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 y = 2x + 1
3x + y = 11
Substitute: 3x + (2x + 1) = 11 → 5x = 10 → x = 2, y = 5
Method 2: Elimination 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 Example: Solve the system 2x + 3y = 12
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 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
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 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).