12x+5=10x−11

To solve the equation, we want to isolate the variable x on one side of the equation.

First, we can simplify both sides by combining like terms.

Starting with the left side:
12x + 5

Next, let's simplify the right side:
10x - 11

Now, we can set the two sides equal to each other:

12x + 5 = 10x - 11

To isolate the variable x, we need to move all terms containing x to one side of the equation.

Let's start by moving the 10x term to the left side by subtracting 10x from both sides:

12x - 10x + 5 = -11

Simplifying, we have:
2x + 5 = -11

Next, we can move the constant term 5 to the right side by subtracting 5 from both sides:

2x + 5 - 5 = -11 - 5

Simplifying, we have:
2x = -16

Finally, to solve for x, we divide both sides of the equation by 2:

(2x)/2 = -16/2

Simplifying, we have:
x = -8

Therefore, the solution to the equation 12x + 5 = 10x - 11 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.(1 point)

To represent the number of days it would take for the two songs to have the same number of listens, let's use the variable x.

For the first song, the number of listens after x days can be represented as: 1,000 + 38x.
(Starting number of listens + new listens per day multiplied by the number of days)

For the second song, the number of listens after x days can be represented as: 900 + 40x.
(Starting number of listens + new listens per day multiplied by the number of days)

We want to find the number of days when the number of listens for both songs is equal. Therefore, we can set the two equations equal to each other:

1,000 + 38x = 900 + 40x

The equation to represent the number of days it would take for the two songs to have the same number of listens is:
1,000 + 38x = 900 + 40x

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.(1 point)

Responses

10x=32+6x ; x=8
10 x equals 32 plus 6 x ; x equals 8

10x=32−6x ; x=2
10 x equals 32 minus 6 x ; x equals 2

10=32+6x ; x=−3
10 equals 32 plus 6 x ; x equals negative 3

10x=32+6x ; x=4

To find the number of visits it would take for the cost of the two plans to be equal, we can set up an equation.

Let x represent the number of visits.

For the first option:
Cost = $10 per visit

For the second option:
Cost = $32 per month + $6 per visit

Setting up the equation:
10x = 32 + 6x

To solve for x, we can subtract 6x from both sides of the equation:
10x - 6x = 32

Simplifying:
4x = 32

To isolate x, we divide both sides of the equation by 4:
(4x)/4 = 32/4

Simplifying:
x = 8

Therefore, 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?(1 point)

The songs will have the same number of listens after
days.

To determine when the two songs will have the same number of listens, let's use the variable x to represent the number of days.

For the first song:
Number of listens = 700 + 52x (Starting number of listens + new listens per day multiplied by the number of days)

For the second song:
Number of listens = 600 + 56x (Starting number of listens + new listens per day multiplied by the number of days)

We want to find when the number of listens for both songs is equal. Therefore, we can set the two equations equal to each other:

700 + 52x = 600 + 56x

To solve for x, we can subtract 600 from both sides of the equation:
700 - 600 + 52x = 600 - 600 + 56x

Simplifying:
100 + 52x = 56x

Next, we can subtract 52x from both sides of the equation to isolate x:
100 + 52x - 52x = 56x - 52x

Simplifying:
100 = 4x

To solve for x, we divide both sides of the equation by 4:
(100) / 4 = (4x) / 4

Simplifying:
25 = x

Therefore, the two songs will have the same number of listens after 25 days.

Miguel’s car has a 20-gallon capacity, and Christina’s car has a 16-gallon capacity. Miguel uses 1.5 gallons of gas per week, and Christina uses 1 gallon of gas per week. When will Miguel and Christina have the same amount of gas in their tanks?(1 point)

Responses

After 8 weeks they will have the same amount of gas in their tanks.
After 8 weeks they will have the same amount of gas in their tanks.

After −8 weeks they will have the same amount of gas in their tanks.
After negative 8 weeks they will have the same amount of gas in their tanks.

After 18 week they will have the same amount of gas in their tanks.
After Start Fraction 1 over 8 End Fraction week they will have the same amount of gas in their tanks.

After 1.6 weeks they will have the same amount of gas in their tanks.
After 1.6 weeks they will have the same amount of gas in their tanks.

To find when Miguel and Christina will have the same amount of gas in their tanks, we can set up an equation.

Let x represent the number of weeks.

For Miguel's car:
Amount of gas used = 1.5 gallons per week

For Christina's car:
Amount of gas used = 1 gallon per week

Setting up the equation:
20 - 1.5x = 16 - 1x

To solve for x, we can subtract 16 from both sides and add 1.5x to both sides:
20 - 16 - 1.5x + 1.5x = 16 - 16 + 1x + 1.5x

Simplifying:
4 = 0.5x

To isolate x, we divide both sides of the equation by 0.5:
(4) / 0.5 = (0.5x) / 0.5

Simplifying:
8 = x

Therefore, Miguel and Christina will have the same amount of gas in their tanks after 8 weeks.

Which of the following equations has exactly one solution?(1 point)

Responses

−3x−8=3x−8
negative 3 x minus 8 equals 3 x minus 8

3x−8=3x+8
3 x minus 8 equals 3 x plus 8

−8x+3=−8x+3
negative 8 x plus 3 equals negative 8 x plus 3

−3x+8=−3x−8

The equation that has exactly one solution is:

−3x + 8 = −3x - 8

Solve for x to determine if the equation has one, none, or infinite solutions.

11x=3(7x−1)−10x(1 point)
Responses

The equation has one solution: x=−3.
The equation has one solution: x equals negative 3 .

The equation has infinite solutions.
The equation has infinite solutions.

The equation has no solution.
The equation has no solution.

The equation has one solution: x=−1.

To solve the equation 11x = 3(7x - 1) - 10x, we can begin by simplifying both sides of the equation.

Starting with the right side:
3(7x - 1) - 10x
21x - 3 - 10x
11x - 3

Now we can rewrite the equation:
11x = 11x - 3

Since 11x is present on both sides of the equation, we can subtract 11x from both sides to isolate the constant term:
11x - 11x = -3 - 11x

Simplifying, we get:
0 = -3 - 11x

Next, we can add 3 to both sides of the equation to eliminate the negative sign:
0 + 3 = -3 + 3 - 11x

Simplifying further:
3 = -11x

Finally, we can divide both sides of the equation by -11 to solve for x:
(3) / -11 = (-11x) / -11

The solution is:
x = -3/11

Therefore, the equation has one solution: x = -3/11.

Solve for x to determine if the equation has one, none, or infinite solutions.

5(2x+2)−4=2x+6(1 point)
Responses

The equation has no solution.
The equation has no solution.

The equation has infinite solutions.
The equation has infinite solutions.

The equation has one solution: x=1.
The equation has one solution: x equals 1 .

The equation has one solution: x=0.
The equation has one solution: x equals 0 .

To solve the equation 5(2x + 2) - 4 = 2x + 6, we can begin by simplifying both sides of the equation.

Starting with the left side:
5(2x + 2) - 4
10x + 10 - 4
10x + 6

Now we can rewrite the equation:
10x + 6 = 2x + 6

To isolate the variable x, we can subtract 2x from both sides:
10x - 2x + 6 = 2x - 2x + 6

Simplifying, we get:
8x + 6 = 6

Next, we can subtract 6 from both sides to isolate the variable:
8x + 6 - 6 = 6 - 6

Simplifying further:
8x = 0

Finally, we can divide both sides of the equation by 8 to solve for x:
8x / 8 = 0 / 8

The solution is:
x = 0

Therefore, the equation has one solution: x = 0.

Solve the following linear equation for x. Then check your answer. Show all your steps for solving and checking.

5x+8+x=32

To solve the linear equation 5x + 8 + x = 32, we can first combine like terms on the left side.

Combining the x terms:
5x + x = 6x

Now we can rewrite the equation:
6x + 8 = 32

Next, we can isolate the variable by subtracting 8 from both sides of the equation:
6x + 8 - 8 = 32 - 8

Simplifying, we have:
6x = 24

To solve for x, we divide both sides of the equation by 6:
(6x)/6 = 24/6

Simplifying, we get:
x = 4

Therefore, the solution to the equation 5x + 8 + x = 32 is x = 4.

To check our answer, we substitute the value of x back into the original equation and verify that both sides are equal.

Original equation:
5x + 8 + x = 32

Substituting x = 4:
5 * 4 + 8 + 4 = 32

Simplifying:
20 + 8 + 4 = 32

Adding:
32 = 32

Since both sides of the equation are equal, our answer of x = 4 is correct.