Question

Solve \(4 x-2(3 x+7)=6+5(x-3)\) A. \(x=\frac{5}{7}\) B. \(x=-\frac{5}{7}\) C. \(x=\frac{23}{7}\) D. \(x=-\frac{7}{5}\)

Short answer

B. \( x = -\frac{5}{7} \)

Step-by-step solution

- Distribute the terms

Distribute the \( -2 \) on the left side of the equation and \( 5 \) on the right side of the equation. \[ 4x - 2(3x + 7) = 6 + 5(x - 3) \] becomes \[ 4x - 6x - 14 = 6 + 5x - 15. \]

- Combine like terms

Combine like terms on both sides of the equation: \[ 4x - 6x - 14 = 6 + 5x - 15 \] simplifies to \[ -2x - 14 = 5x - 9. \]

- Isolate the variable term

Move all terms involving \( x \) to one side of the equation and constant terms to the other side: \[ -2x - 5x = -9 + 14. \]

- Simplify

Simplify both sides of the equation: \[ -7x = 5. \]

- Solve for \( x \)

Divide both sides by \( -7 \): \[ x = \frac{5}{-7} \] or \[ x = -\frac{5}{7}. \]

Key concepts

distributive property

When solving linear equations, the distributive property is a helpful tool. It allows us to multiply a single term by each term inside a set of parentheses. For our problem, we start with: \[ 4x - 2(3x + 7) = 6 + 5(x - 3) \] To distribute, multiply \(-2\) by each term inside its parentheses \((3x and 7)\), and do the same for \(5\) on the right side:

• \(-2 * 3x = -6x\)
• \(-2 * 7 = -14\)
• \(5 * x = 5x\)
• \(5 * -3 = -15\)

So, the equation becomes: \[ 4x - 6x - 14 = 6 + 5x - 15 \] This property helps simplify complex expressions, making the solving process smoother.

combining like terms

The next step involves combining like terms. Like terms are terms that have the same variable raised to the same power. In our example, after distribution, we have the equation: \[ 4x - 6x - 14 = 6 + 5x - 15 \]

• Combine \(4x\) and \(-6x\), which gives \(-2x\)
• On the right side, combine the constants \(6\) and \(-15\), giving \(-9\)

So, it simplifies to: \[ -2x - 14 = 5x - 9 \] Combining like terms reduces the number of terms we need to work with, making it easier to isolate the variable.

isolating the variable

Isolating the variable involves getting the variable term on one side of the equation and the constants on the other. From the previous step, we have: \[ -2x - 14 = 5x - 9 \]

• Move \(5x\) by subtracting it from both sides: \( -2x - 5x - 14 = -9 \)
• Move the constant \(-14\) by adding it to both sides:\( -7x = -9 + 14 \)

This gives us: \[ -7x = 5 \] Our goal is to have \(x\) on one side and all constants on the other. This helps isolate the variable for the solving process.

simplification

The final step is to simplify the equation and solve for \(x\). From: \[ -7x = 5 \]

• Divide both sides by \(-7\):\( x = \frac{5}{-7} \)
• So, the final solution is:\( x = -\frac{5}{7} \)

Simplification ensures that we have the simplest form of the variable and any constants. Therefore, \( x = -\frac{5}{7} \) is our solution. Simplifying the equation in this way helps us to verify our final answer and ensures accuracy.