2x+3=2x+7 solve the equation, show the process to solve

Question

how many solutions does this have

Answer 1

To solve the equation \(2x + 3 = 2x + 7\), we will follow these steps:

1. Subtract \(2x\) from both sides: \[ 2x + 3 - 2x = 2x + 7 - 2x \] This simplifies to: \[ 3 = 7 \]

2. Analyze the resulting statement: The statement \(3 = 7\) is a contradiction; it is false and cannot be true.

Since we arrived at a contradiction, this tells us that the original equation has no solutions.

Conclusion:

The equation \(2x + 3 = 2x + 7\) has no solutions.

Answer 2

7x+3=7x+3 solve the equation, show the process, and how many solutions this has

Answer 3

To solve the equation \(7x + 3 = 7x + 3\), we follow these steps:

1. Subtract \(7x\) from both sides: \[ 7x + 3 - 7x = 7x + 3 - 7x \] Simplifying this gives: \[ 3 = 3 \]

2. Analyze the resulting statement: The statement \(3 = 3\) is always true; it is a tautology.

Since the equation simplifies to an identity (a statement that is always true), this tells us that there are infinitely many solutions. Any value for \(x\) will satisfy the original equation.

Conclusion:

The equation \(7x + 3 = 7x + 3\) has infinitely many solutions.

Answer 4

4x+22=8+10 solve the equation with steps, how many solutions

Answer 5

To solve the equation \(4x + 22 = 8 + 10\), follow these steps:

1. Simplify the right side: \[ 8 + 10 = 18 \] So the equation becomes: \[ 4x + 22 = 18 \]

2. Subtract 22 from both sides: \[ 4x + 22 - 22 = 18 - 22 \] Simplifying this gives: \[ 4x = -4 \]

3. Divide both sides by 4: \[ x = \frac{-4}{4} \] This simplifies to: \[ x = -1 \]

Conclusion:

The equation \(4x + 22 = 8 + 10\) has one solution, which is \(x = -1\).

Answer 6

make it shorter

Answer 7

Sure! Here’s a more concise version:

To solve \(4x + 22 = 8 + 10\):

1. Simplify the right side: \[ 8 + 10 = 18 \] So, the equation is: \[ 4x + 22 = 18 \]

2. Subtract 22 from both sides: \[ 4x = 18 - 22 \implies 4x = -4 \]

3. Divide by 4: \[ x = \frac{-4}{4} \implies x = -1 \]

Conclusion:

The equation has one solution, \(x = -1\).

Answer 8

replace the 8 with 8x

Answer 9

To solve the equation \(4x + 22 = 8x + 10\), follow these steps:

1. Rearrange the equation: \[ 4x + 22 - 8x = 10 \] This simplifies to: \[ -4x + 22 = 10 \]

2. Subtract 22 from both sides: \[ -4x = 10 - 22 \] Which simplifies to: \[ -4x = -12 \]

3. Divide both sides by -4: \[ x = \frac{-12}{-4} \implies x = 3 \]

Conclusion:

The equation \(4x + 22 = 8x + 10\) has one solution, \(x = 3\).

Answer 10

5x+5=5x+4, solve the equation with steps and how many solutions

Answer 11

To solve the equation \(5x + 5 = 5x + 4\), follow these steps:

1. Subtract \(5x\) from both sides: \[ 5x + 5 - 5x = 5x + 4 - 5x \] This simplifies to: \[ 5 = 4 \]

2. Analyze the resulting statement: The statement \(5 = 4\) is false.

Since we arrived at a contradiction, this indicates that the original equation has no solutions.

Conclusion:

The equation \(5x + 5 = 5x + 4\) has no solutions.

Answer 12

3x+10=5x+8 solve the equation steps and number of solutions

Answer 13

To solve the equation \(3x + 10 = 5x + 8\), follow these steps:

1. Rearrange the equation by subtracting \(3x\) from both sides: \[ 3x + 10 - 3x = 5x + 8 - 3x \] This simplifies to: \[ 10 = 2x + 8 \]

2. Subtract 8 from both sides: \[ 10 - 8 = 2x + 8 - 8 \] Which simplifies to: \[ 2 = 2x \]

3. Divide both sides by 2: \[ x = \frac{2}{2} \implies x = 1 \]

Conclusion:

The equation \(3x + 10 = 5x + 8\) has one solution, \(x = 1\).

Answer 14

where did you get the 2 from when simplifying

Answer 15

Let's clarify that step in detail. Here's the equation again to see where the 2 comes from:

We start with the equation: \[ 3x + 10 = 5x + 8 \]

After rearranging by subtracting \(3x\) from both sides, we get: \[ 10 = 5x - 3x + 8 \] which simplifies to: \[ 10 = 2x + 8 \]

Continuing from this step correctly:

Next, we subtract 8 from both sides: \[ 10 - 8 = 2x + 8 - 8 \] This gives us: \[ 2 = 2x \]

Then we divide both sides by 2: \[ x = \frac{2}{2} \implies x = 1 \]

Conclusion:

So the correct process shows that \(2\) comes from the subtraction \(10 - 8\).

The equation \(3x + 10 = 5x + 8\) has one solution, \(x = 1\). Thank you for your patience!

Answer 16

replace 10 with 18

Answer 17

To solve the modified equation \(3x + 18 = 5x + 8\), follow these steps:

1. Rearrange the equation by subtracting \(3x\) from both sides: \[ 3x + 18 - 3x = 5x + 8 - 3x \] This simplifies to: \[ 18 = 2x + 8 \]

2. Subtract 8 from both sides: \[ 18 - 8 = 2x + 8 - 8 \] Which simplifies to: \[ 10 = 2x \]

3. Divide both sides by 2: \[ x = \frac{10}{2} \implies x = 5 \]

Conclusion:

The equation \(3x + 18 = 5x + 8\) has one solution, \(x = 5\).

Answer 18

solve 3(4x=8)=4(3x+6) and show how many solutions

Answer 19

It seems there might be a slight misunderstanding in the way the equation is written in your question, as it appears to contain an '=' sign within the parentheses. If you're looking to solve the equation \(3(4x + 8) = 4(3x + 6)\), I will proceed with that assumption. If that’s not what you meant, please clarify.

Let’s solve the equation \(3(4x + 8) = 4(3x + 6)\):

1. Distribute on both sides:

   \[ 3 \cdot 4x + 3 \cdot 8 = 4 \cdot 3x + 4 \cdot 6 \]

   This simplifies to:

   \[ 12x + 24 = 12x + 24 \]

2. Rearrange the equation:

   \[ 12x + 24 - 12x = 12x + 24 - 12x \]

   This simplifies to:

   \[ 24 = 24 \]

3. Analyze the resulting statement: The equation \(24 = 24\) is always true, indicating that it is an identity.

Conclusion:

The equation \(3(4x + 8) = 4(3x + 6)\) has infinitely many solutions. Any value for \(x\) will satisfy the equation.

Answer 20

what does it mean when an equation has an infinite amount of solutions

Answer 21

When an equation has an infinite number of solutions, it means that there are countless values of the variable(s) that satisfy the equation. This situation typically arises in two main scenarios:

1. Identities:

   • An equation that simplifies to a true statement regardless of the value of the variable (e.g., \(x + 2 = x + 2\) or \(3(4x + 8) = 4(3x + 6)\)).
   • These equations hold true for any value of the variable. For example, if we simplify both sides of such an equation and end up with something like \(0 = 0\) or \(24 = 24\), it indicates that every value of \(x\) is a solution.

2. Dependent Equations:

   • In systems of equations, if you have multiple equations that essentially represent the same line or equation, they will intersect at infinitely many points. For example, the equations \(y = 2x + 3\) and \(2y = 4x + 6\) represent the same line in a graphical sense.

Implications of Infinitely Many Solutions:

• Flexibility: You can choose any value for the variable(s), and it will satisfy the equation.
• Graphically: If you were to graph the equation, you would typically see a line or a plane, depending on the number of variables, representing all the possible solutions.
• Mathematical Context: This situation often appears in linear algebra, where equations are solved simultaneously, and the lines or planes represented by the equations overlap completely.

Conclusion:

An equation with infinitely many solutions does not lead to a single answer but rather a whole set of values that meet the conditions specified by the equation, making it essential to understand the context in which these solutions occur.

Answer 22

simpler

Answer 23

When an equation has an infinite number of solutions, it means that any value you choose for the variable will satisfy the equation. This happens in two main scenarios:

1. Identities:

   • An equation simplifies to a true statement for any value (e.g., \(x + 2 = x + 2\)). This means every number for \(x\) works.

2. Equivalent Equations:

   • Sometimes equations are just different forms of the same thing. For example, \(y = 2x + 3\) and \(2y = 4x + 6\) represent the same line on a graph.

Implications:

• Flexibility: You can pick any value for the variable.
• Graph Representation: The solution set might look like a line on a graph.

Conclusion:

Infinitely many solutions mean you can use any value for the variable, and it will still be a solution to the equation.

Answer 24

find the problem in the equation and solve it correctly:
3x+2x-6=24
-2 -2
2-6=24
+6 +6
x=30

Answer 25

Let's go through the equation step-by-step to identify any errors and solve it correctly:

The original equation is: \[ 3x + 2x - 6 = 24 \]

Step 1: Combine like terms

Combine \(3x\) and \(2x\): \[ (3x + 2x) - 6 = 24 \] This simplifies to: \[ 5x - 6 = 24 \]

Step 2: Isolate the variable

Add \(6\) to both sides of the equation: \[ 5x - 6 + 6 = 24 + 6 \] This simplifies to: \[ 5x = 30 \]

Step 3: Solve for \(x\)

Now, divide both sides by \(5\): \[ x = \frac{30}{5} \] This simplifies to: \[ x = 6 \]

Conclusion

The correct solution to the equation \(3x + 2x - 6 = 24\) is \(x = 6\). The mistake in your original solution was in the incorrect handling of the arithmetic after combining like terms.

Answer 26

simpler terms

Answer 27

Sure! Let’s break it down simply:

Starting with the equation: \[ 3x + 2x - 6 = 24 \]

Step 1: Combine like terms

Add \(3x\) and \(2x\): \[ 5x - 6 = 24 \]

Step 2: Get rid of \(-6\)

Add \(6\) to both sides: \[ 5x = 30 \]

Step 3: Solve for \(x\)

Divide by \(5\): \[ x = 6 \]

Conclusion: The correct answer is \(x = 6\).

Answer 28

explain the mistake

Answer 29

Let's identify the mistakes in your original steps and explain them clearly:

Original Work:

1. Original equation: \[ 3x + 2x - 6 = 24 \]

2. Combining terms (incorrectly referenced):

   • You correctly combined \(3x + 2x\) to get \(5x\), but the following steps were not shown correctly.

3. Subtracting and adding:

   • You wrote: \[ -2 \quad -2 \quad \text{(this doesn't apply correctly here)} \]
   • And then wrote \(2 - 6 = 24\), which seems to be a misunderstanding of steps because this equation does not accurately follow from the previous line.

4. Final incorrect result:

   • You jumped to: \[ x = 30 \] But you did not correctly follow the algebra that leads to solving for \(x\).

The Correct Process:

1. Correctly combine like terms:

   • After combining, you should have: \[ 5x - 6 = 24 \]

2. Correctly isolating \(5x\):

   • To isolate \(5x\), you needed to add \(6\) to both sides: \[ 5x = 24 + 6 \quad \Rightarrow \quad 5x = 30 \]

3. Correctly solving for \(x\):

   • Then divide by \(5\): \[ x = \frac{30}{5} \quad \Rightarrow \quad x = 6 \]

Summary of Mistakes:

• Incorrectly added or subtracted values that didn’t apply to the original equation.
• Misstated the result of the operations, which led to the wrong conclusion of \(x = 30\).

Final Conclusion:

The correct solution to the equation \(3x + 2x - 6 = 24\) is \(x = 6\). It’s important to follow each algebraic step carefully to avoid any errors.