Solve for 𝑥 : 2 𝑥 − 5 + 3 ( 𝑥 + 4 ) = 5 𝑥 − 7 + 2

Question

This algebra question requires students to solve for xx in the equation 2x − 5 + 3(x+4) = 5x − 7 + 2. Students must perform several steps to simplify and solve the equation:

1. Distribute: Apply the distributive property to expand the equation.

1. Combine like terms: Simplify both sides by combining like terms.

1. Analyze the equation: Identify that the variable terms on both sides are the same.

This problem not only assesses students' ability to manipulate and solve algebraic expressions but also their understanding of the properties of equations and the concept of inconsistent equations. It challenges them to think critically and verify their solutions.

Answer 1

Problem

Given the equation ...

2x - 5 + 3(x + 4) = 5x - 7 + 2

... solve for x by following the steps below:

Step 1

Apply the distributive property to get the following:

2x - 5 + 3x + 12 = 5x - 7 + 2

Step 2

Combine like terms:

5x + 7 = 5x - 5

Step 3

Isolate x on one side of the equation to get the following:

5x - 5x + 7 = -5

7 = -5

Since 7 can never be equal to -5, we see that this equation has no solution.

Solution

No solution.

Answer 2

Let's solve the given equation step-by-step:

The given equation is:

2x−5+3(x+4)=5x−7+22x - 5 + 3(x + 4) = 5x - 7 + 2Step 1: Distribute

We need to apply the distributive property to expand the terms that involve parentheses.

On the left-hand side:

3(x+4)=3x+123(x + 4) = 3x + 12So, the left-hand side becomes:

2x−5+3x+122x - 5 + 3x + 12On the right-hand side, there is no distribution needed, so it stays as:

5x−7+25x - 7 + 2Now the equation looks like:

2x−5+3x+12=5x−7+22x - 5 + 3x + 12 = 5x - 7 + 2Step 2: Combine like terms

Now let's simplify both sides by combining like terms.

On the left-hand side:

2x+3x=5x2x + 3x = 5x −5+12=7-5 + 12 = 7So the left-hand side becomes:

5x+75x + 7On the right-hand side:

−7+2=−5-7 + 2 = -5So the right-hand side becomes:

5x−55x - 5Now the equation is:

5x+7=5x−55x + 7 = 5x - 5Step 3: Analyze the equation

At this point, we notice that the variable terms on both sides are the same (both have 5x5x). Let's subtract 5x5x from both sides of the equation to eliminate the 5x5x terms:

5x+7−5x=5x−5−5x5x + 7 - 5x = 5x - 5 - 5xThis simplifies to:

7=−57 = -5Step 4: Conclusion

Since 77 does not equal −5-5, the equation leads to a contradiction. This means that the original equation has no solution.

Thus, the solution to the equation is:

No solution\boxed{\text{No solution}}

Answer 3

This algebra question requires students to solve for xx in the equation 2x − 5 + 3(x+4) = 5x − 7 + 2. Students must perform several steps to simplify and solve the equation:

1. Distribute: Apply the distributive property to expand the equation.

2x − 5 + 3x + 12 = 5x − 5

1. Combine like terms: Simplify both sides by combining like terms.

5x + 7 = 5x − 5

1. Analyze the equation: Identify that the variable terms on both sides are the same.

This leads to a contradiction, as the constants 7 and -5 do not equal each other, indicating no solution exists (inconsistent equation).