Help for Algebra (Solving)

Solving Equations

These equations can be solved relatively easy and without any formal method. But, as you use equations to solve more complex problems, you will want an easier way to solve them.
Pretend you have a scale like the one shown. On the right side there are 45 pennies and on the left side are 23 pennies and an unknown amount of pennies. The scale is balanced, therefore, we know that there must be an equal amount of weight on each side.
As long as the same operation (addition, subtraction, multiplication, etc.) is done to both sides of the scale, it will remain balanced. To find the unknown amount of pennies of the left side, remove 23 pennies from each side of the scale. This action keeps the scale balanced and isolates the unknown amount. Since the weight(amount of pennies) on both sides of the scale are still equal and the unknown amount is alone, we now know that the unknown amount of pennies on the left side is the same as the remaining amount (22 pennies) on the right side.Because an equation represents a scale, it can also be manipulated like one. The diagram below shows a simple equation and the steps to solving it.

Initial Equation / Problem	x + 23	=	45
Subtract 23 from each side	x + 23 - 23	=	45 - 23
Result / Answer	x	=	22

The diagram below shows a more complex equation. This equation has both a constant and a variable on each side. Again, to solve this you must keep both sides of the equation equal; perform the same operation on each side to get the variable "x" alone. The steps to solving the equation are shown below.

Initial Equation / Problem:	x + 23	=	2x + 45
Subtract x from each side	x - x + 23	=	2x - x + 45
Result	23	=	x + 45
Subtract 45 from each side	23 - 45	=	x + 45 - 45
Result	-22	=	x
Answer	x	=	-22Take a look at the equation below. As you can see, after the variable is subtracted from the left and the constants are subtracted from the right, you are still left with 2x on one side. Initial Equation / Problem x + 23 = 3x + 45   Subtract x from each side x - x + 23 = 3x - x + 45 Result 23 = 2x + 45     Subtract 45 from each side 23 - 45 = 2x + 45 - 45 Result -22 = 2x   Switch the left and right sides of the equation 2x = -22 This means that the unknown number multiplied by two, equals -22. To find the value of x, use the process "dividing by the coefficient" described on the next page. Identifying and Using Coefficients The coefficient of a variable is the number which the variable is being multiplied by. In this equation, 2 is the coefficient of x because 2x is present in the equation. Some additional examples of coefficients: Term Coefficient of x 2x 2 0.24x 0.24 x 1 -x -1 Note that in the last two examples, the following rules are applied If the variable has no visible coefficient, then it has an implied coefficient of 1. If the variable only has a negative sign, then it has an implied coefficient of -1. Continue to the next page to see how we use the coefficient of the variable x in the equation, 2, to find the value of x. Using Division Recall beginning to solve the equation "x + 23 = 3x + 45". Applying addition and subtraction gave (from previous page) 2x = -22 But our end goal is to determine what x is, not what 2x is! Imagine that three investors own an equal share in the company Example.Com. The total worth of Example.com is $300,000. To determine what the share of each investor is, simply divide the total investment by 3: $300,000 / 3 = $100,000 Thus, each investor has a $100,000 stake in Example.com. We apply the same idea to finding the value of x. However, instead of dividing by the number of investors, we divide by the coefficient of the variable. Since we determined that the coefficient of x is 2, we divide each side of the equation by 2: After dividing by 2 1x = -11   Finally rewritten as x = -11 Continue to the next page for additional resources and tools for help with basic equations.