Problem 45 - Prealgebra

Question

Solve the equation. \(-7.57-2.42(x+5.54)=6.95\)

Step-by-Step Solution

Answer

The solution is approximately \\(x = -11.54\\).

Step 1: Distribute the Coefficient

First, we distribute the coefficient \(-2.42\) to both terms inside the parentheses \(x + 5.54\). This yields \(-2.42 \, x \, - 2.42 \, \times \, 5.54\). Calculating gives \(-2.42x \,- 13.3988\) approximately.

Step 2: Simplify the Equation

Rewrite the equation with the distributed values: \(-7.57 - 2.42x - 13.3988 = 6.95\). Then, combine the constant terms on the left side: \(-7.57 - 13.3988 = -20.9688\). The equation now looks like \(-20.9688 - 2.42x = 6.95\).

Step 3: Isolate the Variable Term

Add \(+20.9688\) to both sides of the equation to move the constants to the right side. This results in \(-2.42x = 6.95 + 20.9688\), which simplifies to \(-2.42x = 27.9188\).

Step 4: Solve for x

To find \(x\), divide both sides by \(-2.42\): \(x = \frac{27.9188}{-2.42}\). Calculate this to find \(x \approx -11.53842975\).

Step 5: Round the Answer

Round \(x\) to a reasonable decimal place based on initial data precision. Here, \(x \approx -11.54\), rounded to two decimal places.

Key Concepts

Distributive Property

In solving equations, the distributive property is a crucial technique, especially when the equation involves parentheses. The distributive property allows you to multiply a single term outside the parentheses by each term inside the parentheses individually.

For example, in the equation \(-7.57 - 2.42(x + 5.54) = 6.95\), we apply the distributive property to deal with the expression inside parentheses. We multiply \(-2.42\) by \(x\) and \(5.54\). This results in \(-2.42x\) and \(-13.3988\) (since \(-2.42 \times 5.54 = -13.3988\)).

Using the distributive property simplifies the process by eliminating parentheses and expressing the equation in a simpler form.

• Always ensure each term inside the parentheses is multiplied by the term outside.
• This method helps in dealing with equations where operations need to be linked directly to each element inside the parentheses.

Isolating Variables

Isolating variables is an essential strategy in solving equations. The goal is to get the variable alone on one side of the equation. This helps in identifying the value of the variable.

After applying the distributive property, our equation was \(-20.9688 - 2.42x = 6.95\). The next step is to bring all constants on one side, leaving the term with the variable on the other. You do this by undoing addition or subtraction.
You add \(20.9688\) to both sides in this step, which gives \(-2.42x = 27.9188\).

Here, you gradually isolate the variable by eliminating other terms using inverse operations. This breakthrough makes it easier to solve for the variable in further steps.

• Move constants by adding or subtracting to clear them from the variable's side.
• Ensure that whatever operation you do to one side, you also do to the other to maintain balance.

Combining Like Terms

Combining like terms can help simplify an equation, making it easier to solve. This process involves handling terms that contain the same variable raised to the same power and constants.

In the step \(-7.57 - 2.42x - 13.3988 = 6.95\), like terms for constants on the same side were combined, resulting in \(-20.9688 - 2.42x = 6.95\). By simplifying these terms, you ensure the equation is in its simplest form, which makes further steps easier to manage.

Combining like terms reduces the clutter of complex expressions and avoids the confusion of dealing with too many numbers separately.

• Look for terms that have the same variable.
• Also, look to combine constants where possible to simplify early on.