For what value of N would the following equation have no solution?
3(4x−7)+12=2(5x−3)+N(x−3)
Simplify both sides of the equation as much as possible, and solve for x in the equation in terms of N:
3(4x−7)+12=2(5x−3)+N(x−3)
3⋅4x−3⋅7+12=2⋅5x−2⋅3+N⋅x−N⋅3
12x−21+12=10x−6+Nx−3N
12x−9=(10+N)x+(−6−3N)
12x−(10+N)x=(−6−3N)+9
(2−N)x=3−3N
x has exactly one solution unless the denominator is 0  that is, N=2. We make sure that this value renders no solution by substituting:
3(4x−7)+12=2(5x−3)+N(x−3)
3(4x−7)+12=2(5x−3)+2(x−3)
12x−21+12=10x−6+2x−6
12x−9=12x−12
−9=−12
The equation has no solution, and N=2 is the correct answer.
Solve for n:
n+2=−14−n
Explanation:
n+2=−14−n
n+n=−14−2
2n=−16
n=−8
Solve for x: −6x−20=−2x+4(1−3x)
Explanation:
−6x−20=−2x+4(1−3x
−6x−20=−2x+4−12x
−6x−20=−14x+4
−6x+14x=4+20
8x=24
x=3
Solve for b: −14+6b+7−2b=1+5b
−14+6b+7−2b=1+5b
−7+4b=1+5b
4b−5b=1+7
−b=8
b=−8
What is the midpoint coordinate of (1,4) and (7,10)?
Midpoint formula:
What is the midpoint coordinate of (1,2) and (5,2)?
Midpoint formula:
What is the midpoint coordinate of (−2,−1) and (−8,7)?
Midpoint formula:
Solve the following equation:
2x−5+16=30.
We start by isolating the absolute value expression:
2x−5+16=30⇔2x−5=30−16=14⇔x−5=7
This gives us two cases when we remove the absolute value:
x−5=7 and x−5=−7
Then we solve for each case:
x−5=7⇒x=7+5⇒x=12
x−5=−7⇒x=−7+5⇒x=−2
Solve for N:
5(N−6)−2(N+4)=7(N+5)
5(N−6)−2(N+4)=7(N+5)
(5N−30)−(2N+8)=7N+35
5N−30−2N−8=7N+35
3N−38=7N+35
3N−38+38=7N+35+38
3N=7N+73
3N−7N=7N+73−7N
−4N=73
N= 73/4
Solve for x:
(4x+7)+2(x+15)=3(x−17)
(4x+7)+2(x+15)=3(x−17)
(4x+7)+(2x+30)=3x−51
6x+37=3x−51
6x+37−3x=3x−51−3x
3x+37=−51
3x=−51−37
3x=−88
x= 88/3
Which of the following equations has the set of all real numbers as its solution set?
The right side of each equation is 8(N+3), which simplifies by way of distribution to
8(N+3)=8⋅N+8⋅3=8N+24
If the left side of the equation simplifies to an identical expression, the equation has all real numbers as its solutions.
We test the left side of each equation:
2(N+4)+6N=8(N+3)
2(N+4)+6N=2⋅N+2⋅4+6N=2N+8+6N=8N+8
3(N+4)+5N=8(N+3)
3(N+4)+5N=3⋅N+3⋅4+5N=3N+12+5N=8N+12
4(N+4)+4N=8(N+3)
4(N+4)+4N=4⋅N+4⋅4+4N=4N+16+4N=8N+16
5(N+4)+3N=8(N+3)
5(N+4)+3N=5⋅N+5⋅4+3N=5N+20+3N=8N+20
6(N+4)+2N=8(N+3)
6(N+4)+2N=6⋅N+6⋅4+2N=6N+24+2N=8N+24
Of the given choices,
6(N+4)+2N=8(N+3)
can be rewritten as
8N+24=8N+24,
which is an identity and has the set of all real numbers as its solution set.
Consider the incomplete equation
What number replaces the box in order to form an equation with no solution?
Set A to be the number that replaces the box.
Simplify first:
10(N−18)+5N=A(N−12)
10N−180+5N=A(N−12)
15N−180=AN−12A
15N−180+180−AN = AN−12A+180−AN
15N−AN=180−12A
Now solve for N in terms of A:
(15−A)N=180−12A
The only possible value of A that might preclude the existence of a solution is A=15, since it makes the denominator 0. However, let us test this value in the original equation:
10(N−18)+5N=15(N−12)
15N−180=15N−180
As it turns out, replacing the box with 15 yields an identity, not a contradiction, so the solution set is the set of all real numbers. There is no number that fits the description.
Consider the incomplete equation
Which of the following numbers can replace the box to form an equation whose one and only solution is 2?
Rewrite this equation as
4x+17=6(Ax−16)−4x
If 2 is a solution of the equation, then we can substitute 2 for x to make a true arithmetic equation. Replace x with 2 and solve for A:
4⋅2+17=6(A⋅2−16)−4⋅2
8+17=6(2A−16)−8
25 = 12 A  96  8
25=12A−104
129=12A
A=129/12
This number replaces the box in order to form the equation
Solve the following equation for x
We proceed as follows
6x+28=72+4x (Multiply both sides by 4. Remember to distribute the 4 to both summands on both sides.)
6x=44+4x (Subtract 28 from both sides)
2x=44 (Subtract 4x from both sides)
x=22 (Divide both sides by 2)
Define a function f as follows:
f(x)=8x−35
If f(N)=47, evaluate N.
Since f(N)=47, we can plug N in for x and 47 in for f(N) to get the following equation,
8N−35=47
From here, we want to solve for N therefore we must isolate N on one side of the equation and all other numbers on the other side.
8N=47+35
8N=82
N=82÷8
N=10.25
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