Solve for x:
12x + 5 = 10x − 11.
To solve for x, we need to isolate x on one side of the equation.
12x + 5 = 10x − 11
First, move all the terms with x to one side by subtracting 10x from both sides:
12x + 5 - 10x = 10x - 11 - 10x
2x + 5 = -11
Now, isolate x by subtracting 5 from both sides:
2x + 5 - 5 = -11 - 5
2x = -16
Finally, divide by 2 to solve for x:
2x / 2 = -16 / 2
x = -8
Therefore, the solution to the equation is x = -8.
At the beginning of the month, a song on a streaming service already had 1,000 listens, and was getting 38 new listens each day. Meanwhile, a different song already had 900 listens, and was getting 40 new listens each day. Write an equation to show the number of days it would take for the two songs to have the same number of listens. Use x as your variable.
Let's denote the number of days x it would take for the two songs to have the same number of listens.
For the first song with 1,000 listens and 38 new listens each day, the total number of listens after x days would be 1000 + 38x.
For the second song with 900 listens and 40 new listens each day, the total number of listens after x days would be 900 + 40x.
Now, we can write the equation:
1000 + 38x = 900 + 40x
This equation shows that the total number of listens for the first song after x days is equal to the total number of listens for the second song after x days.
An ice skating rink has two ways you can use the facility. The first option is to pay $10 per visit. The second option is to pay $32 per month, and $6 per visit. Write and solve an equation to show the number of visits it would take for the cost of the two plans to be equal.
10x=32+6x; x=8
10=32+6x; x=−3
10x=32+6x; x=4
10x=32−6x; x=2
Let's denote x as the number of visits.
For the first option, the cost is $10 per visit, so the total cost would be 10x.
For the second option, the cost is $32 per month plus $6 per visit, so the total cost would be 32 + 6x.
To find the number of visits it would take for the cost of the two plans to be equal, we set up the equation:
10x = 32 + 6x
Subtracting 6x from both sides gives:
10x - 6x = 32
4x = 32
Divide both sides by 4:
x = 32 / 4
x = 8
So, it would take 8 visits for the cost of the two plans to be equal.
At the beginning of the month, a song on a streaming service had 700 listens, and it was getting 52 new listens each day. Meanwhile, a different song had 600 listens, and it was getting 56 new listens each day. When will the two songs have the same number of listens?
Let's let x be the number of days it will take for the two songs to have the same number of listens.
After x days, the first song with 700 listens and 52 new listens each day will have a total number of listens: 700 + 52x.
After x days, the second song with 600 listens and 56 new listens each day will have a total number of listens: 600 + 56x.
We want to find when these two totals will be equal.
Therefore, we set 700 + 52x = 600 + 56x.
To solve for x:
700 + 52x = 600 + 56x
Substitute in the values
52x - 56x = 600 - 700
-4x = -100
Divide by -4
x = 25
So after 25 days, the two songs will have the same number of listens.
Which of the following equations has no solutions?
A. 6x+3=−6x+3
B. −6x+4=6x+3
C. −6x+4=−6x+3
D. 6x+4=6x+4
The equation has no solutions when the variables can be eliminated to arrive at a false statement. Let's analyze each equation:
A. 6x + 3 = -6x + 3
If we subtract 3 from both sides and add 6x to both sides, the equation becomes:
6x = -6x
This simplifies to:
12x = 0
x = 0
This equation has a solution.
B. -6x + 4 = 6x + 3
If we subtract 4 from both sides and add 6x to both sides, we get:
-6x = 6x - 1
This simplifies to:
0 = 12x + 1
This has a solution.
C. -6x + 4 = -6x + 3
If we subtract -4 from both sides, we get a false statement:
0 = -1
This equation has no solution.
D. 6x + 4 = 6x + 4
If we subtract 4 from both sides, the equation simplifies to:
6x = 6x
This statement is always true.
Therefore, the equation has an infinite number of solutions.
The equation that has no solutions is C. -6x + 4 = -6x + 3.
Which of the following equations has exactly one solution?
A. −3x+8=−3x−8
B. −3x−8=3x−8
C. −8x+3=−8x+3
D. 3x−8=3x+8