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Chapter 10: Problem 61

Solve each equation. $$45=\frac{1}{2}(4 x+30)$$

Short Answer

Expert verified
The solution is \( x = 15 \).

Step by step solution

01

Clear the Fraction

First, we need to eliminate the fraction by multiplying both sides of the equation by 2. This will result in:\[ 2 imes 45 = 2 imes \frac{1}{2}(4x + 30) \]\[ 90 = 4x + 30 \]
02

Eliminate the Constant on One Side

Next, subtract 30 from both sides of the equation to isolate the term with \(x\):\[ 90 - 30 = 4x + 30 - 30 \]\[ 60 = 4x \]
03

Solve for x

Now, divide both sides by 4 to solve for the variable \(x\):\[ \frac{60}{4} = \frac{4x}{4} \]\[ 15 = x \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Solving Equations
Solving equations is all about finding the value of the unknown variable that makes the equation true. Let's break it down step by step.

An equation is like a balanced scale. What you do on one side, you must do on the other. In our exercise, we need to maintain balance while solving for our mystery number, known as a variable. Here, the variable is represented by \( x \).

The process begins by eliminating any fractions (if present) and simplifying both sides to isolate the variable. The end goal is to have the variable by itself on one side and the solution on the other. This is how we solve for the variable and determine its value.
Fractions in Equations
Fractions can complicate equations, but understanding how to handle them makes solving much easier.

First, identify whether the equation contains a fraction. In this case, it is \( \frac{1}{2}(4x + 30) \). The fraction involves multiplying part of the equation, so naturally, our first step is to clear it by multiplying both sides by the denominator (2 in this situation).

  • Multiply every term by the denominator. This cancels out the fraction, simplifying the equation. For example, multiplying both sides of the equation by 2 converts \( \frac{1}{2}(4x + 30) \) to \( 4x + 30 \).
  • It's important to ensure multiplication is applied evenly across all terms involved to maintain equation balance.
  • The goal is to transform the equation so that we can proceed to simpler operations and isolate our variable.
Isolating Variables
Isolating the variable involves all the operations necessary to get the variable \( x \) by itself. Let's see how to do it effectively.

In the previous steps, we cleared the fraction and simplified the equation to \( 90 = 4x + 30 \). We now focus on isolating \( x \):
  • We eliminate the constant on the side of the variable. Here, by subtracting 30 from both sides, we maintain the balance: \( 60 = 4x \).
  • Next, to have \( x \) by itself, divide every term by the coefficient of \( x \). Since our equation is \( 60 = 4x \), divide by 4: \( 15 = x \).
  • This result tells us that when \( x = 15 \), the initial equation is satisfied, completing our solution.
Through each of these steps, keeping the equation balanced is key to correctly finding \( x \).

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Most popular questions from this chapter

A paper company ships reams of paper in a box that weighs 1.3 pounds. Each ream of paper weighs 4.4 pounds, and a box can carry no more than 12 reams of paper. Which inequality best describes the total weight in pounds \(w\) to be shipped in terms of the number of reams of paper \(r\) in each box? \(\mathbf{F} \quad w \geq 1.3+4.4 r, r \geq 12\) \(\mathbf{G} \quad w=1.3+4.4 r, r \leq 12\) \(\mathbf{H} \quad w \leq 1.3+4.4 r, r \leq 12\) \(\mathrm{J} \quad \quad w=1.3+4.4 r, r \geq 12\)

Solve each equation. $$x+3=\frac{1}{2}[(4 x+6)-10]$$

Find the distance between each pair of points. $$E(1,7), F(3,4)$$

A circle has diameter \(d\), radius \(r,\) circumference \(C\), and area \(A .\) Which ratio does not belong? \(\frac{C}{2 r}\), \(\frac{A}{r^{2}}\), \(\frac{C}{r}\), \(\frac{C}{d}\)

Circle \(E\) is inscribed in rhombus \(A B C D\). The diagonals of the rhombus are 10 centimeters and 24 centimeters long. To the nearest tenth centimeter, how long is the radius of circle \(E ?\) (Hint: Draw an altitude from \(E\).)
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