For a variable, or unknown, solving in mathematics, you need to isolate the variable. Isolating one variable means that one it alone on one side of the equal sign. If two or more variables are available, first need the variables to isolate, separate one from them.
Simplifying Equations
The first step in evaluating an equation is to simplify it. Divide both sides by any common factors. For example, if you are simplifying the equation 10xy^2 = 6y, you would divide each side by 2y. The equation then becomes 5xy = 3. Without having a second equation, you cannot solve for "x" or "y"; you can only get one variable in terms of the other. To isolate the "x" variable, divide each side by 5y. Thus, x = 3/(5y). Alternatively, divide each side by 5x to isolate the "y" variable. Thus, y = 3/(5x).
Fractions
If fractions are present in the equation, reduce the fractions. To do this, divide the numerator (top expression) and denominator (bottom expression) by any common factors. For example, if you are simplifying the fraction 10xy^2/(5xy) = 3, you would divide the numerator and denominator by 5xy. The equation then becomes 2y = 3. Note that you do not divide "3" by 5xy. Because the equation 2y = 3 only has one variable, it can be solved. Dividing both sides by 2, you get y = 3/2. That is true regardless of the value of x, and so you do not have enough information to solve for x. The only limitation on the value of x is that it cannot equal zero because the denominator of a fraction cannot equal zero.
Quadratics
Quadratics are equations with 2 as the highest exponent. Quadratics can be factored but each factor will still contain both variables. For example, a^2 + 2ab + b^2 = 0 can be factored into (a + b)^2 = 0. Taking the square root of both sides of the equation, you get a + b = 0. Thus, a = -b and b = -a. As described above, you cannot solve for either variable; you can only get one variable in terms of the other. If you have a second equation, you would substitute "-b" in for "a" and solve that equation for b. You would then use the relationship of a = -b to solve for a.
Trigonometry
One of the most useful trigonometric identities (or expressions that relate trigonometric functions to each other) is (sin x)^2 + (cos x )^2 = 1. That is true for any angle "x". Thus, if an expression contains two angles, you may be able to eliminate one of them by substituting "1" for (sin x)^2 + (cos x )^2. For example, (sin y)[(sin x)^2 + (cos x )^2] = 0.105 simplifies to (sin y)(1) = 0.105, so sin y = 0.105. You can solve for y by taking the inverse sine of 0.105 on a scientific calculator.
Simplifying Equations
The first step in evaluating an equation is to simplify it. Divide both sides by any common factors. For example, if you are simplifying the equation 10xy^2 = 6y, you would divide each side by 2y. The equation then becomes 5xy = 3. Without having a second equation, you cannot solve for "x" or "y"; you can only get one variable in terms of the other. To isolate the "x" variable, divide each side by 5y. Thus, x = 3/(5y). Alternatively, divide each side by 5x to isolate the "y" variable. Thus, y = 3/(5x).
Fractions
If fractions are present in the equation, reduce the fractions. To do this, divide the numerator (top expression) and denominator (bottom expression) by any common factors. For example, if you are simplifying the fraction 10xy^2/(5xy) = 3, you would divide the numerator and denominator by 5xy. The equation then becomes 2y = 3. Note that you do not divide "3" by 5xy. Because the equation 2y = 3 only has one variable, it can be solved. Dividing both sides by 2, you get y = 3/2. That is true regardless of the value of x, and so you do not have enough information to solve for x. The only limitation on the value of x is that it cannot equal zero because the denominator of a fraction cannot equal zero.
Quadratics
Quadratics are equations with 2 as the highest exponent. Quadratics can be factored but each factor will still contain both variables. For example, a^2 + 2ab + b^2 = 0 can be factored into (a + b)^2 = 0. Taking the square root of both sides of the equation, you get a + b = 0. Thus, a = -b and b = -a. As described above, you cannot solve for either variable; you can only get one variable in terms of the other. If you have a second equation, you would substitute "-b" in for "a" and solve that equation for b. You would then use the relationship of a = -b to solve for a.
Trigonometry
One of the most useful trigonometric identities (or expressions that relate trigonometric functions to each other) is (sin x)^2 + (cos x )^2 = 1. That is true for any angle "x". Thus, if an expression contains two angles, you may be able to eliminate one of them by substituting "1" for (sin x)^2 + (cos x )^2. For example, (sin y)[(sin x)^2 + (cos x )^2] = 0.105 simplifies to (sin y)(1) = 0.105, so sin y = 0.105. You can solve for y by taking the inverse sine of 0.105 on a scientific calculator.
No comments:
Post a Comment