Problem 46 - Prealgebra

Question

Solve the equation. \(5.9-0.5(x+5.8)=12.15\)

Step-by-Step Solution

Answer

The solution to the equation is \, \(x = -18.3\).

Step 1: Distribute

Distribute the \, \(-0.5\) \, through the parentheses in the equation \, \(5.9 - 0.5(x + 5.8) = 12.15\). The expression \, \( -0.5(x + 5.8)\) \, becomes:\(-0.5 \times x - 0.5 \times 5.8 = -0.5x - 2.9\).So the equation becomes: \[5.9 - 0.5x - 2.9 = 12.15\]

Step 2: Simplify the Equation

Combine like terms in the equation:\(5.9 - 2.9 = 3.0\).So the equation simplifies to:\[3.0 - 0.5x = 12.15\]

Step 3: Isolate the Variable Term

Subtract \, \(3.0\) \, from both sides of the equation to isolate the variable term:\[3.0 - 0.5x - 3.0 = 12.15 - 3.0\]This simplifies to:\[-0.5x = 9.15\]

Step 4: Solve for the Variable

To solve for \, \(x\), divide both sides of the equation by \, \(-0.5\):\[x = \frac{9.15}{-0.5}\]Calculating the division gives:\[x = -18.3\]

Step 5: Verify the Solution

To ensure the solution is correct, substitute \, \(x = -18.3\) \, back into the original equation and check that both sides are equal.Original equation:\[5.9 - 0.5(x + 5.8) = 12.15\]Substitute \, \(x = -18.3\):\[5.9 - 0.5(-18.3 + 5.8) = 5.9 - 0.5(-12.5)\]\[5.9 + 6.25 = 12.15\]Since left side equals right side \, \(12.15 = 12.15\), the solution is verified.

Key Concepts

Distributive Property

The distributive property is a key concept when dealing with equations, especially those involving parentheses. It allows us to multiply each term inside the parentheses by a factor that is outside the parentheses. For the equation given:

• We have the expression \(-0.5(x + 5.8)\).
• Using the distributive property, we can transform it into \(-0.5 \times x - 0.5 \times 5.8\) which simplifies to \(-0.5x - 2.9\).

This process helps in lifting the equation to a form where the terms can be combined or simply managed on a single line. Think of it as a way to "open up" expressions so you can deal with each term separately.

Isolating Variables

Isolating a variable means rearranging the equation so that the variable you are solving for is by itself on one side of the equation. This step is crucial to find the value of the variable. After distributing and combining like terms, our equation becomes:\[3.0 - 0.5x = 12.15\]To isolate \(x\), follow these steps:

• Subtract 3.0 from both sides to eliminate the constant from the left side: \(3.0 - 0.5x - 3.0 = 12.15 - 3.0\).
• What remains is \(-0.5x = 9.15\).

The goal here is to manipulate the equation so that \(x\) is isolated, and we can then easily find its value.

Combining Like Terms

Combining like terms is a method used to simplify equations by reducing the number of terms. In our problem, this step is necessary right after using the distributive property. The equation transforms to:\[5.9 - 0.5x - 2.9 = 12.15\]Here, you can combine the constant terms:

• The terms \(5.9 \ - 2.9\) both do not contain the variable \(x\).
• Combining these gives \(3.0\).

Thus, the equation simplifies to:\[3.0 - 0.5x = 12.15\]This simplification makes the equation less cluttered and easier to solve.

Verify Solution

Verifying your solution is an essential final step to ensure that the solution found satisfies the original equation. This requires substituting the obtained value back into the equation and checking whether both sides are equal. For our solution, \(x = -18.3\), re-substituting into the original equation:\[5.9 - 0.5(x + 5.8) = 12.15\]Become:\[5.9 - 0.5(-18.3 + 5.8)\]This simplifies to:\[5.9 - 0.5(-12.5) = 5.9 + 6.25 = 12.15\]Because \12.15 = 12.15\, the solution \(x = -18.3\) is verified! It's a great practice to always double-check your solutions in this way.