Chapter Closure
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Exercise 139 Page 385
Gather all of the variable terms on one side of the equation and all of the constant terms on the other side.
Does either of the equations have an isolated variable in it?
x=- 17/3
(- 42,- 18)
We want to solve the given equation. We can start by gathering all of the variable terms on one side and all of the constant terms on the other side using cross products, the Distributive Property, and the Properties of Equality.
x+3/2=x-1/5
CrossMult
Cross multiply
(x+3)(5)=2(x-1)
Distr
Distribute 5
x(5)+3(5)=2(x-1)
Distr
Distribute 2
x(5)+3(5)=2(x)-2(1)
Multiply
Multiply
5x+15=2x-2
SubEqn
LHS-2x=RHS-2x
3x+15=- 2
SubEqn
LHS-15=RHS-15
3x=- 17
DivEqn
.LHS /3.=.RHS /3.
3x/3=- 17/3
SimpQuot
Simplify quotient
x=- 17/3
MoveNegNumToFrac
Put minus sign in front of fraction
x=- 17/3
Checking Our Answer
Checking Our AnswerWe can check our answer by substituting - 173 for x and simplifying. If we get a true statement, our answer is correct. Let's do it!
Our solution is correct because the left-hand side is equal to the right-hand side.
x+3/2=x-1/5
Substitute
x= - 17/3
- 173+3/2? =- 173-1/5
▼
Simplify
NumberToFrac
a = 3* a/3
- 173+ 93/2? =- 173-1/5
OneToFrac
Rewrite 1 as 3/3
- 173+ 93/2? =- 173- 33/5
MoveNegFracToNum
Put minus sign in numerator
- 173+ 93/2? =- 173- 33/5
AddSubFrac
Add and subtract fractions
- 17+93/2? =- 17-33/5
AddSubTerms
Add and subtract terms
- 83/2? =- 203/5
MoveNegNumToFrac
Put minus sign in front of fraction
- 83/2? =- 203/5
MoveNegNumToFrac
Put minus sign in front of fraction
- 83/2? =- 203/5
DivFracD
a/c/b= a/b* c
- 8/2* 3? =- 20/5* 3
Multiply
Multiply
- 8/6? =- 20/15
ReduceFrac
a/b=.a /2./.b /2.
- 8÷ 2/6÷ 2? =- 20/15
ReduceFrac
a/b=.a /5./.b /5.
- 8÷ 2/6÷ 2? =- 20÷ 5/15÷ 5
CalcQuot
Calculate quotient
- 4/3=- 4/3 ✓
In this system of equations, notice that both equations are already solved for y. Let's use the Equal Values Method to set the equations equal to each other and solve for x.
Great! Now, to find the value of y, we need to substitute x=- 42 into either one of the equations in the given system. Let's use the first equation.
The solution, or point of intersection, to this system of equations is the point (- 42,- 18).
y= 12x+3 & (I) y= 13x-4 & (II)
Substitute
(II):y= 1/2x+3
y= 12x+3 12x+3= 13x-4
SubEqn
(II): LHS-1/3x=RHS-1/3x
y= 12x+3 12x+3- 13x=- 4
ExpandFrac
(II): a/b=a * 3/b * 3
y= 12x+3 1* 32* 3x+3- 13x=- 4
ExpandFrac
(II): a/b=a * 2/b * 2
y= 12x+3 1* 32* 3x+3- 1* 23* 2x=- 4
Multiply
(II): Multiply
y= 12x+3 36x+3- 26x=- 4
FactorOut
(II): Factor out x
y= 12x+3 ( 36- 26)x+3=- 4
SubFrac
(II): Subtract fractions
y= 12x+3 3-26x+3=- 4
SubTerm
(II): Subtract term
y= 12x+3 16x+3=- 4
SubEqn
(II): LHS-3=RHS-3
y= 12x+3 16x=- 7
MultEqn
(II): LHS * 6=RHS* 6
y= 12x+3 6( 16x)=6(- 7)
DenomMultFracToNumber
(II): 1 * a/1= a
y= 12x+3 x=6(- 7)
MultPosNeg
(II): a(- b)=- a * b
y= 12x+3 x=- 42
y= 12x+3 x=- 42
Substitute
(I):x= - 42
y= 12( - 42)+3 x=- 42
MultPosNeg
(I): a(- b)=- a * b
y=- 12(42)+3 x=- 42
MoveRightFacToNumOne
(I): 1/b* a = a/b
y=- 422+3 x=- 42
CalcQuot
(I): Calculate quotient
y=- 21+3 x=- 42
AddTerms
(I): Add terms
y=- 18 x=- 42
Checking Our Answer
Let's Check!To check our answer, we will substitute our solution into both equations. If doing so results in true statements, then our solution is correct.
Because both equations are true statements, we know that our solution is correct.
y= 12x+3 & (I) y= 13x-4 & (II)
(I), (II): x= - 42, y= - 18
- 18? = 12( - 42)+3 - 18? = 13( - 42)-4
(I), (II): a(- b)=- a * b
- 18? =- 12(42)+3 - 18? =- 13(42)-4
(I), (II): 1/b* a = a/b
- 18? =- 422+3 - 18? =- 423-4
(I), (II): Calculate quotient
- 18? =- 21+3 - 18? =- 14-4
(I), (II): Add and subtract terms
- 18=- 18 ✓ - 18=- 18 ✓
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