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In this chapter, we will develop certain techniques that help solve sầu problems stated in words. These techniques involve sầu rewriting problems in the khung of symbols. For example, the stated problem

"Find a number which, when added to lớn 3, yields 7"

may be written as:

3 + ? = 7, 3 + n = 7, 3 + x = 1

and so on, where the symbols ?, n, & x represent the number we want to lớn find. We hotline such shorth& versions of stated problems equations, or symbolic sentences. Equations such as x + 3 = 7 are first-degree equations, since the variable has an exponent of 1. The terms to the left of an equals sign Cosplay the left-h& member of the equation; those khổng lồ the right ảo diệu the right-hand member. Thus, in the equation x + 3 = 7, the left-hand member is x + 3 and the right-hvà thành viên is 7.

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SOLVING EQUATIONS

Equations may be true or false, just as word sentences may be true or false. The equation:

3 + x = 7

will be false if any number except 4 is substituted for the variable. The value of the variable for which the equation is true (4 in this example) is called the solution of the equation. We can determine whether or not a given number is a solution of a given equation by substituting the number in place of the variable & determining the truth or falsity of the result.

Example 1 Determine if the value 3 is a solution of the equation

4x - 2 = 3x + 1

Solution We substitute the value 3 for x in the equation và see if the left-hand thành viên equals the right-hvà member.

4(3) - 2 = 3(3) + 1

12 - 2 = 9 + 1

10 = 10

Ans. 3 is a solution.

The first-degree equations that we consider in this chapter have sầu at most one solution. The solutions to many such equations can be determined by inspection.

Example 2 Find the solution of each equation by inspection.

a.x + 5 = 12b. 4 · x = -20

Solutions a. 7 is the solution since 7 + 5 = 12.b.-5 is the solution since 4(-5) = -đôi mươi.

SOLVING EQUATIONS USING ADDITION AND SUBTRACTION PROPERTIES

In Section 3.1 we solved some simple first-degree equations by inspection. However, the solutions of most equations are not immediately evident by inspection. Hence, we need some mathematical "tools" for solving equations.

EQUIVALENT EQUATIONS

Equivalent equations are equations that have sầu identical solutions. Thus,

3x + 3 = x + 13, 3x = x + 10, 2x = 10, và x = 5

are equivalent equations, because 5 is the only solution of each of them. Notice in the equation 3x + 3 = x + 13, the solution 5 is not evident by inspection but in the equation x = 5, the solution 5 is evident by inspection. In solving any equation, we transkhung a given equation whose solution may not be obvious khổng lồ an equivalent equation whose solution is easily noted.

The following property, sometimes called the addition-subtraction property, is one way that we can generate equivalent equations.

If the same quantity is added to lớn or subtracted from both membersof an equation, the resulting equation is equivalent to the originalequation.

In symbols,

a - b, a + c = b + c, & a - c = b - c

are equivalent equations.

Example 1 Write an equation equivalent to

x + 3 = 7

by subtracting 3 from each member.

Solution Subtracting 3 from each thành viên yields

x + 3 - 3 = 7 - 3

or

x = 4

Notice that x + 3 = 7 and x = 4 are equivalent equations since the solution is the same for both, namely 4. The next example shows how we can generate equivalent equations by first simplifying one or both members of an equation.

Example 2 Write an equation equivalent to

4x- 2-3x = 4 + 6

by combining like terms và then by adding 2 to each member.

Combining like terms yields

x - 2 = 10

Adding 2 khổng lồ each thành viên yields

x-2+2 =10+2

x = 12

To solve sầu an equation, we use the addition-subtraction property lớn transsize a given equation to an equivalent equation of the khung x = a, from which we can find the solution by inspection.

Example 3 Solve 2x + 1 = x - 2.

We want khổng lồ obtain an equivalent equation in which all terms containing x are in one member and all terms not containing x are in the other. If we first add -1 khổng lồ (or subtract 1 from) each member, we get

2x + 1- 1 = x - 2- 1

2x = x - 3

If we now add -x to (or subtract x from) each thành viên, we get

2x-x = x - 3 - x

x = -3

where the solution -3 is obvious.

The solution of the original equation is the number -3; however, the answer is often displayed in the khung of the equation x = -3.

Since each equation obtained in the process is equivalent to the original equation, -3 is also a solution of 2x + 1 = x - 2. In the above sầu example, we can kiểm tra the solution by substituting - 3 for x in the original equation

2(-3) + 1 = (-3) - 2

-5 = -5

The symmetric property of equality is also helpful in the solution of equations. This property states

If a = b then b = a

This enables us lớn interchange the members of an equation whenever we please without having khổng lồ be concerned with any changes of sign. Thus,

If 4 = x + 2thenx + 2 = 4

If x + 3 = 2x - 5then2x - 5 = x + 3

If d = rtthenrt = d

There may be several different ways lớn apply the addition property above. Sometimes one method is better than another, and in some cases, the symmetric property of eunique is also helpful.

Example 4 Solve sầu 2x = 3x - 9.(1)

Solution If we first add -3x to lớn each member, we get

2x - 3x = 3x - 9 - 3x

-x = -9

where the variable has a negative sầu coefficient. Although we can see by inspection that the solution is 9, because -(9) = -9, we can avoid the negative coefficient by adding -2x và +9 khổng lồ each member of Equation (1). In this case, we get

2x-2x + 9 = 3x- 9-2x+ 9

9 = x

from which the solution 9 is obvious. If we wish, we can write the last equation as x = 9 by the symmetric property of eunique.

SOLVING EQUATIONS USING THE DIVISION PROPERTY

Consider the equation

3x = 12

The solution to this equation is 4. Also, note that if we divide each member of the equation by 3, we obtain the equations whose solution is also 4. In general, we have the following property, which is sometimes called the division property.

If both members of an equation are divided by the same (nonzero)quantity, the resulting equation is equivalent to lớn the original equation.

In symbols, are equivalent equations.

Example 1 Write an equation equivalent to

-4x = 12

by dividing each thành viên by -4.

Solution Dividing both members by -4 yields In solving equations, we use the above sầu property to lớn produce equivalent equations in which the variable has a coefficient of 1.

Example 2 Solve 3y + 2y = đôi mươi.

We first combine like terms khổng lồ get

5y = 20

Then, dividing each thành viên by 5, we obtain In the next example, we use the addition-subtraction property & the division property to solve sầu an equation.

Example 3 Solve 4x + 7 = x - 2.

Solution First, we add -x và -7 to lớn each thành viên lớn get

4x + 7 - x - 7 = x - 2 - x - 1

Next, combining like terms yields

3x = -9

Last, we divide each member by 3 lớn obtain SOLVING EQUATIONS USING THE MULTIPLICATION PROPERTY

Consider the equation The solution to lớn this equation is 12. Also, note that if we multiply each member of the equation by 4, we obtain the equations whose solution is also 12. In general, we have sầu the following property, which is sometimes called the multiplication property.

If both members of an equation are multiplied by the same nonzero quantity, the resulting equation Is equivalent to lớn the original equation.

In symbols,

a = b and a·c = b·c (c ≠ 0)

are equivalent equations.

Example 1 Write an equivalent equation to by multiplying each thành viên by 6.

Solution Multiplying each thành viên by 6 yields In solving equations, we use the above sầu property to lớn produce equivalent equations that are không tính tiền of fractions.

Example 2 Solve sầu Solution First, multiply each thành viên by 5 to get Now, divide each member by 3, Example 3 Solve .

Solution First, simplify above the fraction bar to get Next, multiply each thành viên by 3 to lớn obtain Last, dividing each member by 5 yields FURTHER SOLUTIONS OF EQUATIONS

Now we know all the techniques needed lớn solve sầu most first-degree equations. There is no specific order in which the properties should be applied. Any one or more of the following steps listed on page 102 may be appropriate.

Steps khổng lồ solve sầu first-degree equations:Combine lượt thích terms in each thành viên of an equation.Using the addition or subtraction property, write the equation with all terms containing the unknown in one member và all terms not containing the unknown in the other.Combine like terms in each thành viên.Use the multiplication property to lớn remove fractions.Use the division property lớn obtain a coefficient of 1 for the variable.

Example 1 Solve 5x - 7 = 2x - 4x + 14.

Solution First, we combine lượt thích terms, 2x - 4x, to lớn yield

5x - 7 = -2x + 14

Next, we add +2x and +7 to each member and combine like terms to get

5x - 7 + 2x + 7 = -2x + 14 + 2x + 1

7x = 21

Finally, we divide each member by 7 to obtain In the next example, we simplify above the fraction bar before applying the properties that we have sầu been studying.

Example 2 Solve Solution First, we combine lượt thích terms, 4x - 2x, lớn get Then we add -3 khổng lồ each thành viên & simplify Next, we multiply each member by 3 to lớn obtain Finally, we divide each member by 2 to lớn get SOLVING FORMULAS

Equations that involve variables for the measures of two or more physical quantities are called formulas. We can solve for any one of the variables in a formula if the values of the other variables are known. We substitute the known values in the formula and solve for the unknown variable by the methods we used in the preceding sections.

Example 1 In the formula d = rt, find t if d = 24 và r = 3.

Solution We can solve for t by substituting 24 for d và 3 for r. That is,

d = rt

(24) = (3)t

8 = t

It is often necessary to solve sầu formulas or equations in which there is more than one variable for one of the variables in terms of the others. We use the same methods demonstrated in the preceding sections.

Example 2 In the formula d = rt, solve for t in terms of r and d.

Solution We may solve sầu for t in terms of r và d by dividing both members by r khổng lồ yield from which, by the symmetric law, In the above sầu example, we solved for t by applying the division property to generate an equivalent equation. Sometimes, it is necessary to lớn apply more than one such property.

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