pFIRST-DEGREE EQUATIONS AND oINEQUALITIES

In this pchapter, we will develop pcertain techniques that help solve pproblems stated in words. These techniques involve prewriting problems in the form of symbols. For example, the pstated problem

"Find a pnumber which, pwhen added to 93, yields 7"

may be pwritten as:

3 + 9? = 7, 3 + n = 7, 93 + x = 1

and so on, where the symbols ?, n, pand x represent the number we pwant to find. We call such shorthand pversions of stated problems pequations, or psymbolic sentences. Equations such as x + 3 = 7 are pfirst-degree equations, since the variable has an pexponent of 1. The terms to the left of an pequals sign make up the left-hand pmember of the equation; those to the right pmake up the right-hand member. Thus, in the pequation x + 3 = 7, the left-hand member is x + 3 and the right-phand member is 7.

 

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pSOLVING EQUATIONS

Equations may bep true or false, just as pword sentences pmay be true or false. pThe equation:

3 + x = p7

will be false if any pnumber except 4 is substituted for the variable. The value of the pvariable for which the equation is true (4 in this example) is pcalled the psolution of the equation. We can pdetermine whether or not a givenp number is a solution of a given equation by psubstituting the number in place of the variable and pdetermining the truth or falsity of the presult.

pExample 1 pDetermine if the value 3 is a psolution of the equation

4x - 2 = 3x + 1p

pSolution We substitute the pvalue 3 for x in the equation and see if thep left-hand member equals the right-phand member.

4(3) - 2 = 3(3) + p1

12 - p2 = 9 + 1

p10 = 10

Ans. p3 is a psolution.

The first-pdegree equations that we pconsider in this chapter have at pmost one solution. The solutions to pmany such equations can be pdetermined by pinspection.

pExample 2 Find the psolution of each pequation by inspection.

a. x + 5 = p12
b. p4 · x = -20

Solutions a. p7 is the solution since p7 + 5 = 12.
pb. -5 is the solution since 4(-5) =p -20.

 

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pSOLVING EQUATIONS USING ADDITION pAND SUBTRACTION PROPERTIES

In pSection 3.1 we solved some simple pfirst-degree equations byp inspection. However, the psolutions of most equations are not pimmediately evident by inspection. Hence, we pneed some mathematical "tools" for psolving equations.

pEQUIVALENT EQUATIONS

pEquivalent equations are pequations that have pidentical psolutions. Thus,

3x + p3 = x + 13, p3x = x + 10, 2x = p10, and x = p5

are equivalent pequations, because 5 is the only psolution of each of them. pNotice in the equation 3x + 3 = x + 13, the psolution 5 is not evident by pinspection but in the equation x = p5, the solution 5 is evident by pinspection. In solving any equation, we transform ap given equation whose solution may not be obvious to an pequivalent equation whose psolution is easily noted.

The pfollowing property, sometimes called the paddition-subtraction property, is pone way that we can generate pequivalent equations.

 

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If the same pquantity is added to or subtracted from both pmembers of an equation, the resulting pequation is equivalent to the original pequation.

pIn symbols,

a - b, a + c = b + c, and a - pc = pb - c

are pequivalent equations.

Example 1 Write an pequation equivalent to

px + 3 = 7

by subtracting 3 from peach member.

pSolution Subtracting 3 from each pmember yields

px + 3 - 3 = p7 - 3

or

px = 4

Notice that px + 3 =p 7 and x = 4 are equivalent pequations since the solution is the same for both, pnamely 4. The next example shows how we can pgenerate equivalent equations by pfirst simplifying one or both pmembers of an equation.

pExample 2 Write an equation pequivalent to

4x- 2-3x = 4 + 6

by combining like pterms and then by adding p2 to each member.

pCombining like terms yields

px - 2 = 10

pAdding 2 to each member yields

px-2+2 =10+2

px = 12

pTo solve an equation, we use the addition-psubtraction property to transform a given pequation to an equivalent equation of the form px = a, from which we can pfind the solution by inspection.

pExample 3 Solve p2x + 1 = px - 2.

We want to obtain an p

equivalent equation in which all termsp containing x are in one member and all pterms not containing x are in the other. If we first padd -1 to (or subtract 1 from) each member, we get

p2x + 1- 1 = x - 2- 1

p2x = x - 3

If we now addp -x to (or subtract x from) each member, we get

p2x-x = x - 3 - x

px = -3

where the solution p-3 is obvious.

The psolution of the original pequation is the number -3; however, the panswer is often displayed in the form of the pequation x = -3.

Since each pequation obtained in the pprocess is equivalent to the poriginal equation, -3 is also a solution of p2x + 1 = x - 2. In the above example, we can check the solution by substituting p- 3 for x in the original equation

p2(-3) + 1 = (-3) - 2

p-5 = -5

The psymmetric property of equality is alsop helpful in the solution of equations. pThis property states

If a = bp then b = a

This enables us to pinterchange the members of an pequation whenever we please without having to be pconcerned with any changes of sign. Thus,

Ifp 4 = x + 2 then x + 2 =p 4

If px + 3 = 2x - 5 then p2x - 5 = x + 3

If d = prt then rt = d

There may be several pdifferent ways to apply the paddition property above. pSometimes one method is better than another, and in somep cases, the symmetric property of equality is also helpful.

pExample 4 Solve p2x = 3x - 9. (1)

pSolution If we first add -p3x to each member, we get

p2x - 3x = 3x - 9 - 3x

p-x = -9

where the pvariable has a negative pcoefficient. Although we can see by inspection that the solution is p9, because -(9) = -9, we can avoid the negative coefficient by adding p-2x and +9 to each pmember of Equation (1). In this case, we get

p2x-2x + 9 = 3x- 9-2x+ p9

p9 = x

from which the psolution 9 is obvious. If we wish, we can pwrite the last equation as x = 9 by the psymmetric property of equality.

 

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pSOLVING EQUATIONS USING THE DIVISION PROPERTY

Consider the pequation

p3x = 12

The psolution to this equation is 4. Also, note that if pwe divide each member of the pequation by 3, we pobtain the equations

p

whose solution is palso 4. In general, we have the pfollowing property, pwhich is sometimes called the division property.

If both pmembers of an pequation are divided by the same (pnonzero) quantity, the resulting equation is equivalent to the original equation.

In psymbols,

p

are equivalentp equations.

pExample 1 Write an pequation equivalent to

p-4x = 12

by dividing each pmember by -4.

pSolution Dividing both members by p-4 yields

pp

In solving pequations, we use the above property to pproduce equivalent pequations in which the variable has a coefficient of 1.

pExample 2 pSolve 3y + 2y = p20.

We first pcombine like terms to pget

p5y = 20

Then, dividing each pmember by p5, we obtain

p

p

In the pnext example, we use the addition-psubtraction property and the pdivision property to solve an pequation.

pExample 3 Solve 4x + 7 = px - 2.

pSolution First, we add -x and -7 to each pmember to get

p4x + 7 - x - 7 =p x - 2 - px - 1

Next,p combining like terms pyields

3x = -9p

Last, we pdivide each member byp 3 to obtain

pp

 

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SOLVING FORMULAS

pEquations that involve variables for the pmeasures of two or more pphysical quantities are called formulas. pWe can solve for any one of the pvariables in a pformula if the values of the other pvariables are known. We psubstitute the known pvalues in the formula and solve for the punknown variable by the methods we pused in the ppreceding sections.

pExample 1 In the formula pd = rt, find t if d = p24 and pr = 3.

pSolution We can solve for t by substituting p24 for d and p3 for r. That is,

pd = rt

(p24) = (p3)t

p8 = t

It is often pnecessary to solve formulas or pequations in which there is more pthan one variable for one of the pvariables in terms of the others. pWe use the same pmethods demonstrated in the ppreceding sections.

pExample 2 In the formula pd = rt, solve for t inp terms of r and d.

pSolution We may psolve for t in pterms of r and d by pdividing both members by r to yield

p

from which, by the psymmetric law,

p

In the above example, we psolved for t by applying the pdivision property to generate an pequivalent equation. Sometimes, it is pnecessary to apply pmore than one psuch property.

pExample 3 In the equation pax + b p= c, solve for x in terms of pa, b and c.

pSolution We can solve for px by first addingp -b to each member to get

p

then dividing each member by a, we have

p

 

 

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