Lesson HOW TO SOLVE EQUATIONS
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an equation has 2 sides to it and an equal sign in the middle that is telling you that the value of the expression on the left side of the equal sign is the same as, or equivalent to, the value of the expression on the right side of the equal sign. an expression is what is on each side of the equation. this is equal to one or more terms. one term is a collection of constants and/or variables that are multiplying or dividing each other. multiple terms are connected to each other by addition or subtraction. some examples of terms are: 5 5x 5x^2 3y etc. some examples of multiple terms are: 5x + 5 5x^2 - 5x + 5 5x + 3y - 5 SOME EXAMPLES OF EQUATIONS 5x = 3 5x^2 + 7x - 5 = 0 x + y = 13 y = 5x - 7 VARIABLES variables are letters or symbols that are used to represent quantities that can vary. for example: y can represent the number 5 at one time and the number 10 at another time. x can represent the number 1 at one time and the number 2 at another time. DEPENDENT AND INDEPENDENT VARIABLES if the value of one variable doesn't depend on the value of another variable, then it is called the independent variable. if the value of one variable depends on the value of another variable, then it is called the dependent variable. the dependent variable and the independent variable are tied to each other through the operations indicated by the equation. an example: the values of 1 and 2 are assigned to the value of x. x is the independent variable because it's value doesn't depend on the value of any other variable. you assign the value to that variable, not the equation. an equation is created that determines the relationship between x and y. for example, the equation is: y = 5*x this equation states that the value of y will always be 5 times the value of x. when x = 1, the value of y will be equal to 5 * 1 = 5 when x = 2, the value of y will be equal to 5 * 2 = 10 the value of x was assigned independently and did not depend on the value of y because you were the one who assigned the values to x. this makes it the independent variable. the value of y was determined by the value of x after the operations indicated in the equation were performed. this make it the dependent variable. the dependent variable in an equation is the variable that you are solving for. it is the variable that has been isolated to one side of the equation when everything else has been isolated to the other side of the equation. in fact, when you solve equations, the main goal is to isolate the variable that you are solving for to one side of the equation with everything else on the other side. you do this by following mathematical rules that allow you to manipulate the equation so you can isolate the variable that you want to solve for to one side of the equation and everything else is on the other side of the equation. EQUATION SOLVING RULES the main equation solving rule is: WHATEVER YOU DO TO ONE SIDE OF THE EQUATION, YOU HAVE TO DO TO THE OTHER SIDE OF THE EQUATION IN ORDER TO PRESERVE THE EQUALITY. preserving the equality means that, if the left side of the equation was equal to the right side of the equation before the operation is performed, then the left side of the equation is still equal to the right side of the equation after the operation is performed. preserving the equality is what allows you to manipulate the equation so that you can isolate the variable you want to solve for to one side of the equation with everything else on the other side of the equation. the good news is: AS LONG AS YOU PERFORM THE SAME OPERATION TO BOTH SIDES OF THE EQUATION, THE EQUALITY WILL BE PRESERVED. some examples of this are: 5 = 5 add 7 to both side of the equation to get: 5+7 = 5+7 simplify to get: 12 = 12 the number are different, but the equality is preserved. the left side of the equation is still equal to the right side of the equation. same thing happens when you subtract the same number from both sides of the equation. 5 = 5 5-7 = 5-7 simplify to get: -2 = -2 5 = 5 7*5 = 7*5 simplify to getr: 35 = 35 50 = 50 50/5 = 50/5 simplify to get: 10 = 10 5 = 5 5^2 = 5^2 simplify to get: 25 = 25 49 = 49 square root (49) = square root (49) simplify to get: 7 = 7 so, the main rule is: WHATEVER YOU DO TO ONE SIDE OF THE EQUATION, YOU HAVE TO DO TO THE OTHER SIDE OF THE EQUATION IN ORDER TO PRESERVE THE EQUALITY. SOLVING PROBLEMS INVOLVING EQUATIONS the main goal in solving equation is to isolate the variable you are solving for to one side of the equation while the remaining variables and constants are on the other side of the equation. for example: if the original equation is x + y = 32, and you want to solve for x, then you have to isolate x to one side of the equation and move or keep 32 and y on the other side of the equation. you can isolate x to the left side of the equation or to the right side of the equation. it really doesn't matter. most of the times, however, you will want to isolate x to the left side of the equation. keep in mind that you can switch sides in an equality equation and the equality will still be preserved. if the equation is x = y, then the equation y = x means the same thing. whether x is on the left or the right doesn't matter. if you see an equation that says x^2 + 2x = y, then you can flip it around so that it says y = x^2 + 2x and it will mean the same thing. back to our original equation being x + y = 32 we want to solve for x we will want to keep x on the left side of the equation in this case. this means that we want x on the left side of the equation and we want y and 32 on the right side of the equation. REVERSE OPERATIONS ON VARIABLES OR CONSTANTS we move variables or constants from one side of an equation to the other side of an equation by performing reverse operations. if the variable or constant is an addition, we apply a subtraction. if the variable or constant is a subtraction, we apply an addition. if the variable or constant is a multiplication, we apply a division. if the variable or constant is a division, we apply a multiplication. back to our example: our equation is x + y = 32 we subtract y from both sides of the equation to get: x + (y - y) = 32 - y since (y-y) equals 0 equals 0, we are left with: x + 0 = 32 - y, which simplifies to: x = 32 - y x has been isolated to the left side of the equation. y has been moved to the right side of the equation by performing reverse operations. y was being added on the left side of the equation so we subtracted y from both sides of the equation. this had the effect of canceling y out on the left side of the equation because any value subtracted from itself will be equal to 0. now we'll solve some simple problems so you can see how the rule works in practice. EXAMPLE 1 3x + 2y = 3 solve for y subtract 3x from both sides of the equation to get: 3x + 2y - 3x = 3 - 3x reorder the terms to get: 3x - 3x + 2y = 3 - 3x combine like terms to get: 0 + 2y = 3 - 3x simplify to get: 2y = 3 - 3x divide both sides of the equation by 2 to get: 2y / 2 = (3 - 3x) / 2 simplify to get: y = (3 - 3x) / 2 EXAMPLE 2: 5x = 10 solve for x divide both sides of the equation by 5 to get: 5x/5 = 10/5 simplify to get: x = 2 EXAMPLE 3: sqrt(x) = 6 square both sides of the equation to get: x = 6^2 simplify to get: x = 36 EXAMPLE 4: x/5 = 7/3 multiply both sides of the equation by 5 to get: x/5 * 5 = (7/3) * 5 simplify to get: x = (7/3) * 5 simplify further to get: x = 35/3 EXAMPLE 5: 7/3 = 5/x multiply both sides of this equation by x to get: (7/3) * x = (5/x) * x simplify to get: 7x/3 = 5 multiply both sides of this equation by 3 to get: 7x/3 * 3 = 5 * 3 which becomes: 7x = 15 divide both sides of this equation by 7 to get: 7x / 7 = 15 / 7 which becomes: x = 15/7. in all of these equations, you confirm that the answer is correct by replacing the value of x in the original equation with the value of x that you solved for to determine if the original equation is still true. it it is still true, then the variable that you solved for has been solved for successfully. let's do that for one of these examples. we'll use example 5. EXAMPLE 5 the original equation was: 7/3 = 5/x our solution was: x = 15/7 we replace x in the original equation with 15/7 to get: 7/3 = 5 / (15/7) this is equivalent to: 7/3 = 5 * 7/15 simplify this to get: 7/3 = 7/3 the left side of the equation is equal to the right side of the equation. this means that the solution is confirmed to be good. SOME MORE EXAMPLES: EXAMPLE 6: this one's a little more complicated, but it shows you that the same rule applies regardless of the equation. note that ^ indicates exponentiation. x^2 means the same as x squared. the equation is: x^2 + y^2 = 36 you want to solve for y. subtract x^2 from both sides of the equation to get: x^2 - x^2 + y^2 = 36 - x^2 simplify this to get: y^2 = 36 - x^2 take the square root of both sides of the equation to get: y = +/- sqrt(36 - x^2) that's your solution. you have isolated y to one side of the equation and everything else to the other side of the equation. y is the dependent variable. x is the independent variable. you choose the values of x that you want. the value of y depends on the relationship expressed by the equation. for example: when x = 1, y = sqrt(36 - 1^2) which is equal to sqrt (36 - 1) which is equal to sqrt(35) when x = 6, y = sqrt(36 - 6^2) which is equal to sqrt (36 - 36) which is equal to sqrt(0) which is equal to 0. when x = 7, y = sqrt(36 - 7^2) which is equal to sqrt (36 - 49) which is equal to sqrt(-13). sqrt(-13) is not a real number. it is an imaginary number. if you are only looking for real numbers as a solution, then the value of x cannot be greater than 6 because anything greater than 6 will result in an imaginary number as an answer. x is still the independent variable and y is still the dependent variable because you can assign any value to x that you want and the resulting value of y is dependent on that value of x. there are restriction, however, based on whether or not you want a real number to be a solution to the equation. if you do, then the value of x is restricted to only those values of x that will yield a real value of y after substituting x for the value of x in the equation.
