Question

1. andre solved an equation, but when he checked his answer he saw his solution was incorrect. he knows he made a mistake, but he can’t find it. where is andre’s mistake and what is the solution to the equation?

-2(3x - 5) = 4(x + 3) + 8
-6x + 10 = 4x + 12 + 8
-6x + 10 = 4x + 20
10 = -2x + 20
-10 = -2x
5 = x

Explanation:

Step1: Analyze the original equation

The original equation is \(-2(3x - 5)=4(x + 3)+8\). First, we apply the distributive property. For the left - hand side, \(-2\times(3x-5)=-2\times3x-2\times(-5)=-6x + 10\). For the right - hand side, \(4\times(x + 3)+8=4x+12 + 8=4x+20\). So the correct equation after distributing should be \(-6x + 10=4x+20\).

Looking at Andre's steps, in his second step (the line \(-6x + 10=4x + 12+8\) is correct, but then in the next step, he has \(-6x + 10=4x+20\)? Wait, no, looking at the given steps: Wait, the first step of Andre's solution is \(-2(3x - 5)=4(x + 3)+8\), then he expands to \(-6x + 10=4x+12 + 8\) (this is correct, because \(4(x + 3)=4x+12\) and then +8). Then the next step: he wrote \(-6x + 10=4x+20\)? Wait, no, in the given steps, there is a line \(-6x + 10=4x+12 + 8\) and then \(-6x + 10=4x+20\) (this is correct because \(12 + 8 = 20\)). Then the next step: he has \(10=-2x + 20\)? Wait, no, let's re - examine the steps.

Wait, let's solve the equation correctly:

1. Start with \(-2(3x - 5)=4(x + 3)+8\)
2. Distribute: \(-6x+10 = 4x + 12+8\)
3. Simplify the right - hand side: \(-6x + 10=4x+20\)
4. Subtract \(4x\) from both sides: \(-6x-4x + 10=4x-4x+20\), which gives \(-10x + 10=20\)
5. Subtract 10 from both sides: \(-10x+10 - 10=20 - 10\), so \(-10x = 10\)
6. Divide both sides by \(-10\): \(x=\frac{10}{-10}=-1\)

Wait, but looking at Andre's steps:

Andre's steps:

1. \(-2(3x - 5)=4(x + 3)+8\)
2. \(-6x + 10=4x+12 + 8\) (correct)
3. \(-6x + 10=4x+20\) (correct, since \(12 + 8 = 20\))
4. Then he has \(10=-2x + 20\) (this is the mistake! To get from \(-6x + 10=4x+20\) to the next step, we should subtract \(4x\) from both sides: \(-6x-4x+10=4x-4x + 20\), which is \(-10x + 10=20\), not \(10=-2x + 20\). So Andre made a mistake when moving the \(x\) terms. He should have subtracted \(4x\) from both sides, but instead, he seems to have added \(6x\) and subtracted \(4x\) incorrectly.

Now, let's solve the equation correctly:

Starting from \(-2(3x - 5)=4(x + 3)+8\)

Step1: Distribute both sides

Left side: \(-2\times3x-2\times(-5)=-6x + 10\)

Right side: \(4\times x+4\times3 + 8=4x+12 + 8=4x+20\)

So the equation becomes \(-6x + 10=4x+20\)

Step2: Move the \(x\) terms to one side

Subtract \(4x\) from both sides: \(-6x-4x+10=4x-4x + 20\)

\(-10x + 10=20\)

Step3: Move the constant term to the other side

Subtract 10 from both sides: \(-10x+10 - 10=20 - 10\)

\(-10x = 10\)

Step4: Solve for \(x\)

Divide both sides by \(-10\): \(x=\frac{10}{-10}=-1\)

Wait, but maybe I misread the steps. Let's check the given steps again. The steps are:

\(-2(3x - 5)=4(x + 3)+8\)

\(-6x + 10=4x+12 + 8\)

\(-6x + 10=4x+20\)

\(-6x + 10=4x+20\) (wait, no, the next step is \(-6x + 10=4x+20\) then \(-6x + 10=4x+20\)? No, the next step in the given image is \(-6x + 10=4x+20\) then \(10=-2x + 20\)? Wait, no, the steps as per the image:

Looking at the vertical steps:

1. \(-2(3x - 5)=4(x + 3)+8\)

2. \(-6x + 10=4x+12 + 8\)

3. \(-6x + 10=4x+20\)

4. \(10=-2x + 20\) (this is the mistake. From \(-6x + 10=4x+20\), to isolate the \(x\) terms, we should subtract \(4x\) and add \(6x\) or subtract \(4x\) from both sides: \(-6x-4x+10=20\) which is \(-10x + 10=20\), not \(10=-2x + 20\). So Andre's mistake is in the step where he moves the \(x\) terms. He should have \(-10x+10 = 20\) instead of \(10=-2x + 20\).

Now, to find the correct solution:

Start from the correct equation after distributing: \(-6x + 10=4x+20\)

Subtract \(4x\) from both sides: \(-6x-4x+10=4x-4x + 20\)

\(-10x + 10=20\)

Subtract 10 from both sides: \(-10x=20 - 10\)

\(-10x = 10\)

Divide by \(-10\): \(x=-1\)

Wait, but maybe the original equation was written differently. Wait, the first line of the equation: is it \(-2(3x - 5)=4(x + 3)+8\) or \(2(3x - 5)=4(x + 3)+8\)? Wait, the image shows \(-2(3x - 5)=4(x + 3)+8\) (the negative sign is there).

Wait, let's re - do the correct solution:

1. Given \(-2(3x - 5)=4(x + 3)+8\)
2. Distribute: \(-6x + 10=4x+12 + 8\) (because \(4(x + 3)=4x + 12\) and then +8)
3. Simplify RHS: \(-6x + 10=4x+20\)
4. Subtract \(4x\) from both sides: \(-6x-4x+10=4x-4x + 20\) → \(-10x + 10=20\)
5. Subtract 10 from both sides: \(-10x=20 - 10\) → \(-10x = 10\)
6. Divide by \(-10\): \(x=-1\)

Now, looking at Andre's steps, in the step where he goes from \(-6x + 10=4x+20\) to \(10=-2x + 20\), he made a mistake. He should have \(-10x+10 = 20\) instead of \(10=-2x + 20\) (he probably added \(6x\) to both sides: \(-6x+6x + 10=4x+6x+20\) → \(10 = 10x+20\), which is also wrong, or he subtracted \(4x\) and added \(6x\) incorrectly).

To find the correct solution, we solve the equation as follows:

Starting from \(-2(3x - 5)=4(x + 3)+8\)

- Distribute: \(-6x + 10=4x+12 + 8\)
- Simplify RHS: \(-6x + 10=4x+20\)
- Subtract \(4x\) from both sides: \(-10x + 10=20\)
- Subtract 10: \(-10x=10\)
- Divide by \(-10\): \(x = - 1\)

Wait, but maybe the original equation was \(2(3x - 5)=4(x + 3)+8\) (without the negative sign). Let's check that case. If the equation is \(2(3x - 5)=4(x + 3)+8\):

1. Distribute: \(6x-10 = 4x+12 + 8\)
2. Simplify RHS: \(6x-10=4x+20\)
3. Subtract \(4x\): \(2x-10 = 20\)
4. Add 10: \(2x=30\)
5. Divide by 2: \(x = 15\). But the given steps have \(-6x + 10\), so the original equation has a negative sign.

So, the mistake is in the step where Andre moves the \(x\) terms (from \(-6x + 10=4x+20\) to \(10=-2x + 20\) is incorrect; it should be \(-10x+10 = 20\)).

To find the correct solution:

We have \(-6x + 10=4x+20\)

Subtract \(4x\) from both sides: \(-6x-4x+10=4x-4x + 20\)

\(-10x + 10=20\)

Subtract 10 from both sides: \(-10x=20 - 10\)

\(-10x = 10\)

Divide both sides by \(-10\): \(x=-1\)

Answer:

The mistake is in the step where Andre transforms \(-6x + 10 = 4x+20\) to \(10=-2x + 20\) (he should get \(-10x + 10=20\) instead). The correct solution of the equation \(-2(3x - 5)=4(x + 3)+8\) is \(x=-1\)