Question

3x + \frac{4}{5} = 7 - 2x\
what is the solution to the equation?\
x =

Explanation:

Step1: Add 2x to both sides

To isolate the variable terms on one side, we add \(2x\) to both sides of the equation \(3x+\frac{4}{5}=7 - 2x\). This gives us \(3x + 2x+\frac{4}{5}=7-2x + 2x\), which simplifies to \(5x+\frac{4}{5}=7\).

Step2: Subtract \(\frac{4}{5}\) from both sides

Next, we subtract \(\frac{4}{5}\) from both sides to get the term with \(x\) alone on the left. So, \(5x+\frac{4}{5}-\frac{4}{5}=7-\frac{4}{5}\). Calculating the right side: \(7=\frac{35}{5}\), so \(\frac{35}{5}-\frac{4}{5}=\frac{31}{5}\). Thus, \(5x=\frac{31}{5}\).

Step3: Divide both sides by 5

To solve for \(x\), we divide both sides by 5. Remember that dividing by 5 is the same as multiplying by \(\frac{1}{5}\), so \(x=\frac{31}{5}\times\frac{1}{5}=\frac{31}{25}\)? Wait, no, wait. Wait, step 2: \(7-\frac{4}{5}\) is \(\frac{35 - 4}{5}=\frac{31}{5}\)? Wait, no, wait, original equation: \(3x+\frac{4}{5}=7 - 2x\). Let's redo step 1 and step 2 correctly.

Wait, step 1: Add \(2x\) to both sides: \(3x + 2x+\frac{4}{5}=7\), so \(5x+\frac{4}{5}=7\). Then step 2: Subtract \(\frac{4}{5}\) from both sides: \(5x=7-\frac{4}{5}\). Convert 7 to fifths: \(7=\frac{35}{5}\), so \(\frac{35}{5}-\frac{4}{5}=\frac{31}{5}\)? Wait, no, that's wrong. Wait, 7 is \(\frac{35}{5}\), so \(\frac{35}{5}-\frac{4}{5}=\frac{31}{5}\)? Wait, no, 35 - 4 is 31, so \(\frac{31}{5}\). Then step 3: Divide both sides by 5: \(x=\frac{31}{5}\div5=\frac{31}{5}\times\frac{1}{5}=\frac{31}{25}\)? Wait, no, that's incorrect. Wait, no, I made a mistake. Wait, the equation is \(3x+\frac{4}{5}=7 - 2x\). Let's do it again.

Correct step 1: Add \(2x\) to both sides: \(3x + 2x+\frac{4}{5}=7\), so \(5x+\frac{4}{5}=7\). Then step 2: Subtract \(\frac{4}{5}\) from both sides: \(5x=7-\frac{4}{5}\). 7 is \(\frac{35}{5}\), so \(\frac{35}{5}-\frac{4}{5}=\frac{31}{5}\)? Wait, no, 35 - 4 is 31, so \(\frac{31}{5}\). Then step 3: Divide both sides by 5: \(x=\frac{31}{5}\div5=\frac{31}{5}\times\frac{1}{5}=\frac{31}{25}\)? Wait, that can't be. Wait, no, wait, maybe I messed up the sign. Wait, original equation: \(3x+\frac{4}{5}=7 - 2x\). Let's move all x terms to left and constants to right. So \(3x + 2x=7-\frac{4}{5}\). So \(5x=7-\frac{4}{5}\). 7 is \(\frac{35}{5}\), so \(\frac{35}{5}-\frac{4}{5}=\frac{31}{5}\)? Wait, no, 35 - 4 is 31, so \(\frac{31}{5}\). Then \(x=\frac{31}{5}\div5=\frac{31}{25}\)? Wait, that's 1.24. Wait, but let's check with another method. Let's multiply both sides by 5 to eliminate fractions.

Multiply both sides by 5: \(5\times(3x+\frac{4}{5})=5\times(7 - 2x)\). This gives \(15x + 4 = 35 - 10x\). Then add \(10x\) to both sides: \(15x + 10x+4=35\), so \(25x + 4 = 35\). Subtract 4: \(25x=31\). Then divide by 25: \(x=\frac{31}{25}\). Ah, there we go. I see, earlier mistake was in step 3. So when we multiply both sides by 5, we get \(15x + 4 = 35 - 10x\), then add \(10x\): \(25x + 4 = 35\), subtract 4: \(25x = 31\), so \(x=\frac{31}{25}=1.24\).

So let's redo the steps correctly:

Step1: Multiply both sides by 5 (to eliminate fraction)

The equation is \(3x+\frac{4}{5}=7 - 2x\). Multiply each term by 5: \(5\times3x+5\times\frac{4}{5}=5\times7 - 5\times2x\). This simplifies to \(15x + 4 = 35 - 10x\).

Step2: Add 10x to both sides

To get all x terms on the left, add \(10x\) to both sides: \(15x + 10x+4=35 - 10x + 10x\), which gives \(25x + 4 = 35\).

Step3: Subtract 4 from both sides

Subtract 4 from both sides: \(25x + 4 - 4=35 - 4\), so \(25x = 31\).

Step4: Divide both sides by 25

Divide both sides by 25: \(x=\frac{31}{25}\).

Yes, that's correct. So the solution is \(x=\frac{31}{25}\…

Answer:

\(\frac{31}{25}\) (or 1.24)