In this chapter, we will develop certain techniques that help solve 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 3, yields 7"
may be written as:
3 + ? = 7, 3 + n = 7, 3 + x = 1
và so on, where the symbols ?, n, & x represent the number we want lớn find. We Gọi such shorthand 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 khổng lồ the left of an equals sign trang điểm the left-hvà thành viên of the equation; those to the right ảo diệu the right-hvà member. Thus, in the equation x + 3 = 7, the left-h& thành viên is x + 3 and the right-hvà member 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 và 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 and see if the left-hvà thành viên equals the right-hand 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 lớn 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) = -trăng tròn.
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, and 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 transsize 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 or subtracted from both membersof an equation, the resulting equation is equivalent lớn the originalequation.
In symbols,
a - b, a + c = b + c, và 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 member yields
x + 3 - 3 = 7 - 3
or
x = 4
Notice that x + 3 = 7 & 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 lượt thích terms và then by adding 2 khổng lồ each thành viên.
Combining lượt thích terms yields
x - 2 = 10
Adding 2 lớn each thành viên yields
x-2+2 =10+2
x = 12
To solve an equation, we use the addition-subtraction property khổng lồ transform a given equation lớn an equivalent equation of the khung x = a, from which we can find the solution by inspection.
Example 3 Solve sầu 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 lớn (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 member, 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 form of the equation x = -3.
Since each equation obtained in the process is equivalent khổng lồ 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 echất lượng is also helpful in the solution of equations. This property states
If a = b then b = a
This enables us to interchange the members of an equation whenever we please without having to lớn 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 khổng lồ apply the addition property above. Sometimes one method is better than another, & in some cases, the symmetric property of equality is also helpful.
Example 4 Solve 2x = 3x - 9.(1)
Solution If we first add -3x khổng lồ each thành viên, 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 sầu 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 equality.
SOLVING EQUATIONS USING THE DIVISION PROPERTY
Consider the equation
3x = 12
The solution khổng lồ 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.
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If both members of an equation are divided by the same (nonzero)quantity, the resulting equation is equivalent 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 lớn produce equivalent equations in which the variable has a coefficient of 1.
Example 2 Solve 3y + 2y = trăng tròn.
We first combine like terms khổng lồ get
5y = 20
Then, dividing each member by 5, we obtain
In the next example, we use the addition-subtraction property & the division property lớn solve an equation.
Example 3 Solve sầu 4x + 7 = x - 2.
Solution First, we add -x và -7 lớn each member lớn get
4x + 7 - x - 7 = x - 2 - x - 1
Next, combining like terms yields
3x = -9
Last, we divide each member by 3 to lớn obtain
SOLVING EQUATIONS USING THE MULTIPLICATION PROPERTY
Consider the equation
The solution khổng lồ this equation is 12. Also, note that if we multiply each thành viên of the equation by 4, we obtain the equations
whose solution is also 12. In general, we have 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 & 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 member by 6 yields
In solving equations, we use the above property lớn produce equivalent equations that are free of fractions.
Example 2 Solve
Solution First, multiply each thành viên by 5 khổng lồ get
Now, divide each member by 3,
Example 3 Solve
Solution First, simplify above sầu the fraction bar khổng lồ get
Next, multiply each member by 3 khổng lồ obtain
Last, dividing each member by 5 yields
FURTHER SOLUTIONS OF EQUATIONS
Now we know all the techniques needed lớn solve 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 lớn 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 & all terms not containing the unknown in the other.Combine like terms in each member.Use the multiplication property khổng lồ remove sầu fractions.Use the division property to 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, lớn yield
5x - 7 = -2x + 14
Next, we add +2x and +7 khổng lồ each thành viên & combine like terms to lớn get
5x - 7 + 2x + 7 = -2x + 14 + 2x + 1
7x = 21
Finally, we divide each member by 7 khổng lồ obtain
In the next example, we simplify above the fraction bar before applying the properties that we have been studying.
Example 2 Solve sầu
Solution First, we combine like terms, 4x - 2x, to lớn get
Then we add -3 to each member và simplify
Next, we multiply each thành viên by 3 khổng lồ obtain
Finally, we divide each member by 2 khổng lồ get
SOLVING FORMULAS
Equations that involve sầu variables for the measures of two or more physical quantities are called formulas. We can solve sầu 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 & 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 sầu for t by substituting 24 for d và 3 for r. That is,
d = rt
(24) = (3)t
8 = t
It is often necessary lớn 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 sầu for t in terms of r và d.
Solution We may solve for t in terms of r and d by dividing both members by r to lớn yield
from which, by the symmetric law,
In the above sầu example, we solved for t by applying the division property to lớn generate an equivalent equation. Sometimes, it is necessary khổng lồ apply more than one such property.
