How to Solve Equations for the Indicated Variable

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Elementary algebra is one of the main branches of mathematics and introduces the concept of using variables to represent numbers and defines the rules on how to manipulate equations containing these variables. Variables are important because they allow for the formulation of generalized mathematical laws and allow the introduction of unknown numbers into equations. It is these unknown numbers that are the focus when solving equations with variables. These variables are frequently represented as x and y.

Linear and Parabolic Equations

• Move any constant values from the side of the equation with the variable to the other side of the equals sign. For example, for the equation 4x² + 9 = 16, subtract 9 from both sides of the equation to remove the 9 from the variable side: 4x² + 9 - 9 = 16 - 9, which simplifies to 4x² = 7.

• Divide the equation by the coefficient of the variable term. For example, if 4x² = 7, then (4x² / 4) = 7 / 4, which results in x² = 1.75 which becomes x = sqrt(1.75) = 1.32.

• Take the proper root of the equation to remove the exponent of the variable. For example, if x² = 1.75, then sqrt(x²) = sqrt(1.75), which results in x = 1.32.

• Isolate the expression containing the variable by using the appropriate arithmetic method to cancel out the constant on the side of the variable. For example, if sqrt(x + 27) + 11 = 15, by using subtraction: sqrt(x + 27) + 11 - 11 = 15 - 11 = 4.

• Raise both sides of the equation to the power of the root of the variable to rid the variable of the root. For example, sqrt(x + 27) = 4, then sqrt(x + 27)² = 4² and x + 27 = 16.

• Isolate the variable by using the appropriate arithmetic method to cancel out the constant on the side of the variable. For example, if x + 27 = 16, by using subtraction: x = 16 - 27 = -11.

• Set the equation equal to zero. For example, for the equation 2x² - x = 1, subtract 1 from both sides to set the equation to zero: 2x² - x - 1 = 0.

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• Factor or complete the square of the quadratic, whichever is easier. For example, for the equation 2x² - x - 1 = 0, it is easiest to factor so: 2x² - x - 1 = 0 becomes (2x + 1)(x - 1) = 0.

• Solve the equation for the variable. For example, if (2x + 1)(x - 1) = 0, then the equation equals zero when: 2x + 1 = 0 becomes 2x = -1 becomes x = -(1 / 2) or when x - 1 = 0 becomes x = 1. These are the solutions to the quadratic equation.

Equations With Fractions

• Factor each denominator. For example, 1 / (x - 3) + 1 / (x + 3) = 10 / (x² - 9) can be factored to become: 1 / (x - 3) + 1 / (x + 3) = 10 / (x - 3)(x + 3).

• Multiply each side of the equation by the least common multiple of the denominators. The least common multiple is the expression that each denominator can divide evenly into. For the equation 1 / (x - 3) + 1 / (x + 3) = 10 / (x - 3)(x + 3), the least common multiple is (x - 3)(x + 3). So, (x - 3)(x + 3) (1 / (x - 3) + 1 / (x + 3)) = (x - 3)(x + 3)(10 / (x - 3)(x + 3)) becomes (x - 3)(x + 3) / (x - 3) + (x - 3)(x + 3) / (x + 3 = (x - 3)(x + 3)(10 / (x - 3)(x + 3).

• Cancel terms and solve for x. For example, cancelling terms for the equation (x - 3)(x + 3) / (x - 3) + (x - 3)(x + 3) / (x + 3 = (x - 3)(x + 3)(10 / (x - 3)(x + 3) finds: (x + 3) + (x - 3) = 10 becomes 2x = 10 becomes x = 5.

Exponential Equations

• Isolate the exponential expression by cancelling any constant terms. For example, 100(14²) + 6 = 10 becomes 100(14²) + 6 - 6 = 10 - 6 = 4.

• Cancel out the coefficient of the variable by dividing both sides by the coefficient. For example, 100(14²) = 4 becomes 100(14²) / 100 = 4 / 100 = 14² = 0.04.

• Take the natural log of the equation to bring down the exponent containing the variable. For example, 14² = 0.04 becomes: ln(14²)= ln(0.04) = 2xln(14) = ln(1) - ln(25) = 2xln(14) = 0 - ln(25).

• Solve the equation for the variable. For example, 2xln(14) = 0 - ln(25) becomes: x = -ln(25) / 2ln(14) = -0.61.

Logarithmic Equations

• Isolate the natural log of the variable. For example, the equation 2ln(3x) = 4 becomes: ln(3x) = (4 / 2) = 2.

• Convert the log equation to an exponential equation by raising the log to an exponent of the appropriate base. For example, ln(3x) = (4 / 2) = 2 becomes: e^ln(3x) = e².

• Solve the equation for the variable. For example, e^ln(3x) = e² becomes 3x / 3 = e² / 3 becomes x = 2.46.

References

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