For example, to solve the system 3x + 2y = 10 and 2x - 4y = -4, multiply the first equation by 2 to get 6x + 4y = 20, then add the second equation to eliminate y and get 8x = 16.
Solve for x to get x = 2, then substitute x = 2 into either equation to get y = 5. For example, to solve the system x + y = 7 and 2x - y = 1, solve the first equation for y to get y = 7 - x, then substitute 7 - x for y in the second equation to get 2x - (7 - x) = 1. Substitution Method: To solve a system of equations using the substitution method, solve one of the equations for one of the variables, then substitute the expression for that variable into the other equation.For example, to solve x^2 + 6x + 5 = 0, add and subtract (6/2)^2 = 9 to get (x + 3)^2 - 4 = 0, then factor to get (x + 3 - 2)(x + 3 + 2) = 0, which gives x = -5 or x = -1.Ī system of equations is a set of equations with multiple variables. To complete the square, add and subtract (b/2a)^2 to the equation, then factor the resulting trinomial. Completing the Square: Completing the square is a method for solving quadratic equations.Point-Slope Form: The point-slope form of a linear equation is y - y1 = m(x - x1), where m is the slope and (x1, y1) is a point on the line.Ī quadratic equation is an equation of the form ax^2 + bx + c = 0, where a, b, and c are constants, and x is the variable.Slope-Intercept Form: The slope-intercept form of a linear equation is y = mx + b, where m is the slope and b is the y-intercept.
For example, to solve 2x + 3 = 7, subtract 3 from both sides to get 2x = 4, then divide both sides by 2 to get x = 2. Solving Linear Equations: To solve a linear equation for one variable, use inverse operations to isolate the variable on one side of the equation.The formula is x = (-b ± sqrt(b^2 - 4ac)) / 2aĪlgebraic Equations Cheat Sheet Linear EquationsĪ linear equation is an equation of the form Ax + By = C, where A, B, and C are constants, and x and y are variables. Quadratic Formula: The quadratic formula can be used to find the roots of a quadratic equation.FOIL Method: (a + b)(c + d) = ac + ad + bc + bd.Factoring: To factor a polynomial, find the common factor of all the terms and divide each term by that factor.Distributive Property: a(b+c) = ab + ac.Division: To divide two or more terms, just divide the coefficients and subtract the exponents of the like variables.Multiplication: To multiply two or more terms, just multiply the coefficients and add the exponents of the like variables.Subtraction: To subtract two or more terms, just subtract the coefficients of the like terms.Addition: To add two or more terms, just combine the coefficients of the like terms.$True$: $\ c_\ x) \nRightarrowĬall by Name and Call by Value may not reduce to the Normal Form! Call by Name terminates more often than Call by Value.Algebraic Expressions Basic Algebraic Expressions.untyped lambda calculus is turing complete.Function application is left associative $\lambda x.\ f\ x\ y = \lambda x.\ ((f\ x)\ y)$.
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