How to Solve the Linear Equation 5(x + 43) = 2(3x + 4)

We are asked to solve the given linear equation for the variable \(x\). The equation is:

\(5(x + 43) = 2(3x + 4)\)

Step-by-Step Solution to Find the Value of x

To solve this equation, we need to simplify both sides and isolate \(x\). Here are the steps:

1. Distribute the numbers outside the parentheses:
   Apply the distributive property \(a(b+c) = ab + ac\) to both sides of the equation.
   • Left side: \(5 \times x + 5 \times 43\)
   • Right side: \(2 \times 3x + 2 \times 4\)
   This gives us:
   \(5x + 215 = 6x + 8\)
2. Gather the terms with \(x\) on one side and constant terms on the other side:
   We can subtract \(5x\) from both sides to move the \(x\) term to the right side, or subtract \(6x\) to move it to the left. Let's move the \(x\) terms to the right side and the constant terms to the left side.
   Subtract \(8\) from both sides:
   \(5x + 215 - 8 = 6x + 8 - 8\)
   \(5x + 207 = 6x\)
   Subtract \(5x\) from both sides:
   \(5x + 207 - 5x = 6x - 5x\)
   \(207 = x\)
3. Read the solution:
   The equation simplifies to \(207 = x\), which means \(x = 207\).

Verification

To verify the solution, substitute \(x = 207\) back into the original equation:

\(5(x + 43) = 2(3x + 4)\)

Substitute \(x = 207\):

\(5(207 + 43) = 2(3 \times 207 + 4)\)

Simplify inside the parentheses:

\(5(250) = 2(621 + 4)\)

\(1250 = 2(625)\)

\(1250 = 1250\)

Since both sides are equal, our solution \(x = 207\) is correct.

Summary of the Solution Process

Solving linear equations involves simplifying the equation using distributive properties and then isolating the variable by performing the same operations on both sides of the equation.

In this specific case:

• Initial equation: \(5(x + 43) = 2(3x + 4)\)
• After distribution: \(5x + 215 = 6x + 8\)
• After rearranging terms: \(215 - 8 = 6x - 5x\)
• Final solution: \(207 = x\)

Thus, the value of \(x\) that satisfies the equation is \(207\).