Problem 28 - Elementary Algebra Within Reach

Question

Solve the equation and check your solution. (Some of the equations have no solution.) $$3[(5 x+1)-4]=4(2 x-3)$$

Step-by-Step Solution

Answer

The solution to the equation is \( x = -3/7 \).

Step 1: Expand and Simplify the Equation

Expand the equation using the distributive property of multiplication over addition or subtraction: \( 3[(5 x+1)-4]=4(2 x-3) \) becomes \( 3(5x +1-4) = 4(2x -3) \), which simplifies to \( 3(5x -3) = 8x - 12 \).

Step 2: Isolate 'x' and Solve

Rearrange the equation to isolate 'x' on one side. First, distribute '3' on the left side: \( 15x - 9 = 8x - 12 \). After rearranging, you get: \( 15x - 8x = -12 + 9 \), which simplifies to \( 7x = -3 \). Therefore, the value of 'x' is: \( x = -3/7 \).

Step 3: Verify the Solution

Substitute 'x' with -3/7 into the original equation to verify the solution. \( 3[(5*(-3/7)+1)-4]=4(2*(-3/7)-3) \), this simplifies to \( -1 = -1 \), which verifies our solution.

Key Concepts

Distributive Property

The distributive property is a key concept in solving equations, especially when dealing with expressions that involve parentheses. The property allows you to simplify expressions by multiplying a single term outside the parenthesis across each term inside. Here's how it works in our example:

• Start with: \(3[(5x + 1) - 4]\).
• Apply the distributive property: \(3 \times (5x) + 3 \times 1 - 3 \times 4\).
• This simplifies to \(15x + 3 - 12\).
• Further simplify to \(15x - 9\).

By distributing, we transform a complex equation into a simpler form, making it easier to isolate the variable later.

Isolating Variables

Isolating the variable is an important step in solving equations. The goal is to get the variable by itself on one side of the equation to determine its value. This requires performing operations such as addition, subtraction, multiplication, or division on both sides of the equation.In our example, we need to isolate \(x\):

• Combine like terms on both sides: \(15x - 9 = 8x - 12\).
• Move all terms containing \(x\) to one side: \(15x - 8x\).
• This results in \(7x = -12 + 9\).
• Simplify to find \(7x = -3\).
• Solve for \(x\) by dividing both sides by 7: \(x = -\frac{3}{7}\).

By strategically isolating the variable, you uncover the value needed to solve the equation. Remember, whatever you do to one side must be done to the other to keep the equation balanced.

Checking Solutions

Checking solutions is a critical last step in solving equations. This ensures that the calculated value satisfies the original equation, confirming its correctness. You verify the solution by substituting the value of the variable back into the original equation.For instance, substitute \(x = -\frac{3}{7}\) back into the equation:

• Original equation: \(3[(5x + 1) - 4] = 4(2x - 3)\).
• After substitution: \(3[(5\cdot(-\frac{3}{7}) + 1) - 4] = 4(2\cdot(-\frac{3}{7}) - 3)\).
• Simplify both sides: \(-1 = -1\).

Since both sides equal, the solution \(x = -\frac{3}{7}\) is correct. Always perform this step to verify your solution is accurate and free of errors.