Q. Solve the Equation: \( -3x + 1 + 10x = x + 4 \)

Write the original equation.

\[ -3x + 1 + 10x = x + 4 \]

Explanation: We start from the given equation and will simplify both sides step by step.

Combine like terms on the left-hand side.

On the left, the terms involving x are -3x and 10x. Combine them by adding their coefficients:

\[ -3x + 10x = ( -3 + 10 )x = 7x \]

So the equation becomes:

\[ 7x + 1 = x + 4 \]

Explanation: Combining like terms reduces the number of terms and simplifies the equation.

Move the x-term from the right-hand side to the left-hand side.

Subtract x from both sides to collect x-terms on the left:

\[ 7x + 1 – x = x + 4 – x \]

Simplify both sides:

\[ (7x – x) + 1 = 4 \]

\[ 6x + 1 = 4 \]

Explanation: Subtracting x from each side keeps the equality true while isolating x-terms on one side.

Isolate the x-term by removing the constant on the left-hand side.

Subtract 1 from both sides:

\[ 6x + 1 – 1 = 4 – 1 \]

Simplify:

\[ 6x = 3 \]

Explanation: Subtracting the same number from both sides maintains equality and removes the constant term from the x-side.

Solve for x by dividing both sides by the coefficient of x.

Divide both sides by 6:

\[ x = \frac{3}{6} \]

Simplify the fraction:

\[ x = \frac{1}{2} \]

Explanation: Dividing by the coefficient isolates x and gives its numerical value.

Check the solution by substituting back into the original equation.

Substitute x = \frac{1}{2} into the left-hand side:

\[ -3\left(\frac{1}{2}\right) + 1 + 10\left(\frac{1}{2}\right) = -\frac{3}{2} + 1 + 5 = -\frac{3}{2} + \frac{2}{2} + \frac{10}{2} = \frac{9}{2} \]

Substitute x = \frac{1}{2} into the right-hand side:

\[ \frac{1}{2} + 4 = \frac{1}{2} + \frac{8}{2} = \frac{9}{2} \]

Both sides are equal, so the solution is verified.

Final answer.

\[ x = \frac{1}{2} \]