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Chapter 7: Problem 55

$$\text { Solve: } \frac{x+3}{4}-\frac{x+1}{10}=\frac{x-2}{5}-1$$

Short Answer

Expert verified
The solution to the equation is \( x = -11 \)

Step by step solution

01

Clear the Fractions

First, we determine the least common multiple of the denominators 4, 10 and 5 which is 20. Multiply every term of the equation by 20 to eliminate the fractions: \( 20 \cdot \frac{x+3}{4} - 20 \cdot \frac{x+1}{10} = 20 \cdot \frac{x-2}{5} - 20 \cdot 1 \). This simplifies to \( 5x + 15 - 2x - 2 = 4x - 8 - 20 \)
02

Simplify and Collect Like Terms

Combine the like terms on both sides: \( 5x - 2x = 4x - 8 - 20 + 2 + 15 \). This simplifies to \( 3x = 4x - 11 \)
03

Solve for x

Next, we isolate x by subtracting 4x from both sides of the equation: \( 3x - 4x = -11 \). This gives the solution \( x = -11 \)

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Key Concepts

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

Clearing Fractions in Equations
When solving linear equations, fractions can make the process more complex. Clearing fractions is a technique used to simplify such equations. You multiply every term by the least common multiple (LCM) of all the denominators to eliminate the fractions entirely.

For example, consider the equation \( \frac{x+3}{4} - \frac{x+1}{10} = \frac{x-2}{5} -1 \). To clear the fractions, you need to find the LCM of 4, 10, and 5, which is 20. By multiplying each term by 20, you won't have any fractions left, and the equation becomes easier to work with.
Finding Least Common Multiple in Algebra

Understanding LCM


The least common multiple or LCM of two or more numbers is the smallest number that is a multiple of each of the numbers. In algebra, finding the LCM is crucial when you need to combine fractions with different denominators or clear fractions from an equation.

To find the LCM, list the multiples of each number and identify the smallest multiple that appears in each list. Alternatively, you can use the prime factorization method where you multiply the highest power of all prime factors that appear in the numbers.
Simplifying Algebraic Expressions
Simplicity is key to working with algebraic expressions. Simplifying can involve combining like terms, which are terms that have the same variables raised to the same power. You add or subtract the coefficients of these like terms.

For instance, in our example \(5x + 15 - 2x\) becomes \(3x + 15\) after simplifying. Similarly, on the right side, we combine constants and like terms to create a more straightforward expression. The process reduces the equation's complexity and makes the solution more accessible.
Isolating Variables in Equations
The ultimate goal when solving an equation is to isolate the variable, meaning getting the variable on one side and all other terms on the opposite side. This is done by performing the same mathematical operations on both sides of the equation.

In our equation, after simplifying, we end up with \(3x = 4x - 11\). To isolate \(x\), we subtract \(4x\) from both sides resulting in \(x = -11\). This step is fundamental because it gives us the value of the variable which is the solution to the equation.

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Most popular questions from this chapter

a. Graph the solution set of the system: $$\left\\{\begin{array}{c}x \geq 0 \\\y \geq 0 \\\3 x-2 y \leq 6 \\\y \leq-x+7\end{array}\right.$$ b. List the points that form the corners of the graphed region in part (a). c. Evaluate \(2 x+5 y\) at each of the points obtained in \(\operatorname{part}(\mathrm{b})\)

Solve each system by the method of your choice. $$\left\\{\begin{array}{l} (x-1)^{2}+(y+1)^{2}=5 \\ 2 x-y=3 \end{array}\right.$$

Solve the system for \(x\) and \(y\) in terms of \(a_{1}, b_{1}, c_{1}, a_{2}, b_{2}\) and \(c_{2}\): $$\left\\{\begin{array}{l}a_{1} x+b_{1} y=c_{1} \\\a_{2} x+b_{2} y=c_{2}\end{array}\right.$$

Use the formula for the area of a rectangle and the Pythagorean Theorem to solve A small television has a picture with a diagonal measure of 10 inches and a viewing area of 48 square inches. Find the length and width of the screen. (IMAGE CAN NOT COPY)

Graphing utilities can be used to shade regions in the rectangular coordinate system, thereby graphing an inequality in two variables. Read the section of the user's manual for your graphing utility that describes how to shade a region. Then use your graphing utility to graph the inequalities. $$y \leq 4 x+4$$
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