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Results pageAlgebrator HomeSolve Simplify Factor Expand Graph GCF LCM New Example Help All solvers Tutorials Desktop Math Solver Get it on Google Play Get it on Apple Store Solve Simplify Factor Expand Graph GCF LCMNew Example Keyboard Solve √ ∛ e i π s c t l L ≥ ≤ 1 2 3 4 5 6 7 8 9 0 . , ≤ ≥ ^ √ ⬅ ➡ F _ ÷ | ( * / ⌫ A ↻ x y = + - G OKThe Addition Principle After studying this section, you will be able to: 1. Solve equations of the form x + b = c using the addition principle. 2. Using the Addition Principle When we use the equals sign (=), we indicate that two expressions are equal in value. This is called an equation . For example, x + 5 = 23 is an equation. By choosing certain procedures, you can go step by step from a given equation to the equation x = some number. The number is the solution to the equation. One of the first procedures used in solving equations has an application in our everyday world. Suppose that we place a 10 -kilogram box on one side of a seesaw and a 10 -kilogram stone on the other side. If the center of the box is the same distance from the balance point as the center of the stone, we would expect the seesaw to balance. The box and the stone do not look the same, but they have the same value in weight. If we add a 2 -kilogram lead weight to the center of weight of each object at the same time, the seesaw should still balance. The results are equal. There is a similar principle in mathematics. We can state it in words like this. The Addition Principle If the same number is added to both sides of an equation, the results on each side are equal in value. We can restate it in symbols this way. For real numbers a, b, c if a=b thenat+tc=b+ec Here is an example. If 3=6/2 , then 3+5=6/2+5 Since we added the same amount 5 to both sides, each side has an equal value. 3+5=6/2+5 8 =6/2+10/2 8 = 16/2 8=8 We can use the addition principle to solve an equation. EXAMPLE 1 Solve for x . x + 16 = 20 x + 16 + (-16) = 20 + (-16) Use the addition principle to add -16 to both sides. x+0=4 Simplify. x=4 The value of x is 4 . We have just found the solution of the equation. The solution is a value for the variable that makes the equation true. We then say that the value, 4 , in our example, satisfies the equation. We can easily verify that 4 is a solution by substituting this value in the original equation. This step is called checking the solution. Check . x + 16 = 20 4+16 ≟ 20 20 = 20 ✔ When the same value appears on both sides of the equals sign, we call the equation an identity . Because the two sides of the equation in our check have the same value, we know that the original equation has been correctly solved. We have found the solution. When you are trying to solve these types of equations, you notice that you must add a particular number to both sides of the equation. What is the number to choose? Look at the number that is on the same side of the equation with x , that is, the number added to x . Then think of the number that is opposite in sign . This is called the additive inverse of the number. The additive inverse of 16 is -16 . The additive inverse of -3 is 3 . The number to add to both sides of the equation is precisely this additive inverse. It does not matter which side of the equation contains the variable. The x term may be on the right or left. In the next example the x term will be on the right. EXAMPLE 2 Solve for x . 14 =x- 3 14+3=x-3 +3 Add 3 to both sides, since 3 is the additive inverse of -3 . This will eliminate the -3 on the right and isolate x . 17 =x+0 Simplify. 17=x The value of x is 17 . Check . 14 = x-3 14 ≟ 17-3 Replace x by 17 . 14 = 14 ✔ Simplify. It checks. The solution is x = 17 . Before you add a number to both sides, you should always simplify the equation. The following example shows how combining numbers by addition separately, on both sides of the equation—simplifies the equation. EXAMPLE 3 Solve for x . 15 +2=3+x+2 17=x+5 Simplify by adding. 17+ (-5) =x+5+(-5) Add the value -5 to both sides, since -5 is the additive inverse of 5 . 12=x Simplify. The value of x is 12 . Check . 15+2 = 3+x+2 15+2 ≟ 3+12+2 Replace x by 12 in the original equation. 17=17 ✔ It checks. In Example 3 we added -5 to each side. You could subtract 5 from each side and get the same result. In earlier lesson we discussed how subtracting a 5 is the same as adding a negative 5 . Do you see why? We can determine if a value is the solution to an equation by following the same steps used to check an answer. Substitute the value to be tested for the variable in the original equation. We will obtain an identity if the value is the solution. EXAMPLE 4 Is x = 10 the solution to the equation -15 + 2 = x-3 ? If it is not, find the solution. We substitute 10 for x in the equation and see if we obtain an identity. -15+2=x-3 -15+2=10-3 -13 ≠ 7 This is not true. It is not an identity. Thus, x = 10 is not the solution. Now we take the original equation and solve to find the solution. -15+2=x-3 -13=x-3 Simplify by adding. -13+3=x-3+3 Add 3 to both sides. 3 is the additive inverse of -3 . -10=x Check to see if x = -10 is the solution. The value x = 10 was incorrect because of a sign error. We must be especially careful to write the correct sign for each number when solving equations. EXAMPLE 5 Find the value of x that satisfies the equation 1/5+x = − 1/10+1/2 To combine the fractions, the fractions must have common denominators. The least common denominator (LCD) of the fractions is 10 . (1*2)/(5*2)+x = − 1/10+(1*5)/(2*5) Change each fraction to an equivalent fraction with a denominator of 10 . 2/10 + x = − 1/10+5/10 This is an equivalent equation. 2/10+x = 4/10 Simplify by adding. 2/10+(-2/10)+x = 4/10+(-2/10) Add the additive inverse of 2/10 to each side x=2/10 Add the fractions. x= 1/5 Simplify the answer. Check . We substitute 1/5 for x in the original equation and see if we obtain an identity. 1/5+x = − 1/10+1/2 1/5+1/5 ≟ − 1/10+1/2 Substitute 1/5 for x 2/5 ≟ − 1/10+1/2 2/5 = 4/10 2/5 = 2/5 ✔ It checks. The Multiplication Principle After studying this section, you will be able to: 1. Solve equations of the form 1/ax=b . 2. Solve equations of the form ax = b . Solving Equations of the Form 1/ax=b The addition principle allows us to add the same number to both sides of an equation. What would happen if we multiplied each side of an equation by the same number? For example, what would happen if we multiplied each side of an equation by 3 ? To answer this question, let’s return to our simple example of the box and the stone on a balanced seesaw. If we triple the number of weights on each side (we are multiplying each side by 3 ), the seesaw should still balance. The ‘‘weight value’’ of each side remains equal. In words we can state this principle thus. Multiplication Principle If both sides of an equation are multiplied by the same number, the results on each side are equal in value. In symbols we can restate the multiplication principle this way. | For real numbers a,b,c wihc #0 ifa@=b thenca=cb | Let us look at an equation where it would be helpful to multiply each side of the equation by 3 . EXAMPLE 1 Solve for x . 1/3x=-15 We know that (3)(1/3) = 1 . We will multiply each side of the equation by 3 , because we want to isolate the variable x . (3)(1/3x)=3(-15) Multiply each side of the equation by 3 since (3)(1/3) = 1 . (3/1)(1/3)(x)=-45 1x=-45 Simplify. x= -45 Check . 1/3(-45) ≟ -15 Substitute -45 for x in the original equation. -15=-15 ✔ It checks. Note that 1/5x can be written as x/5 . To solve the equation x/5=3 , we could multiply each side of the equation by 5 . Try it. Then check your solution. Solving Equations of the Form ax = b We can see that using the multiplication principle to multiply each side of an equation by 1/2 is the same as dividing each side of the equation by 2 . Thus, it would seem that the multiplication principle would allow us to divide each side of the equation by any nonzero real number. Is there a real-world example...