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Chapter 7: Problem 47

Solve each equation by doing the same thing to both sides. $$ \frac{3 k-5}{5}+k=5+k $$

Short Answer

Expert verified
k = 10

Step by step solution

01

Combine Like Terms

Notice that there is a common term on both sides of the equation, which is k. Subtract k from both sides to simplify: When: $$ \frac{3k-5}{5} + k = 5 + k - k $$, this simplifies to $$ \frac{3k-5}{5} = 5 $$
02

Clear the Fraction

Multiply both sides of the equation by 5 to clear the denominator: $$ 5 \times \frac{3k - 5}{5} = 5 \times 5 $$ Simplifies to: $$ 3k - 5 = 25 $$
03

Isolate the Variable

Add 5 to both sides to isolate the term with the variable k: $$ 3k - 5 + 5 = 25 + 5 $$ This simplifies to: $$ 3k = 30 $$
04

Solve for k

Divide both sides by 3 to solve for k: $$ \frac{3k}{3} = \frac{30}{3} $$ This simplifies to: $$ k = 10 $$

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Key Concepts

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

Combining Like Terms
When you start solving a linear equation, you will often find similar terms on both sides of the equation.
These are called 'like terms'. Combining like terms means simplifying the equation by merging these similar terms together.
In the given exercise, the term 'k' appears on both sides. To simplify, you subtract 'k' from both sides:
$$\frac{3k-5}{5} + k = 5 + k - k$$
This will simplify the equation to:
$$\frac{3k-5}{5} = 5$$.
Clearing Fractions
Fractions can make equations look complicated, but there's a simple trick to clear them.
Multiplying every term in the equation by the denominator removes the fraction.
In the exercise:
$$5 \times \frac{3k - 5}{5} = 5 \times 5$$
By doing this, the fraction is eliminated, and the equation becomes:
$$3k - 5 = 25$$.
Always remember to balance the equation by multiplying or dividing both sides equally.
Isolating the Variable
After clearing fractions, the next goal is to isolate the variable you're solving for.
This usually requires moving other terms to the opposite side of the equation.
In our exercise, to isolate 'k', you add 5 to both sides:
$$3k - 5 + 5 = 25 + 5$$.
Simplifying this, you get:
$$3k = 30$$.
It's crucial to do the same mathematical operation on both sides to keep the equation balanced.
Algebraic Manipulation
Finally, you need to perform some algebraic manipulation to solve for the variable.
This step may involve division or multiplication to isolate the variable completely.
In the given equation, divide both sides by 3:
$$\frac{3k}{3} = \frac{30}{3}$$
This simplifies to:
$$k = 10$$.
Always double-check your work to confirm that your solution satisfies the original equation.
Practice is key to mastering these steps in solving linear equations.

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Most popular questions from this chapter

Factor the expression on the left side of each equation as much as possible, and find all the possible solutions. It will help to remember that \(x^{4}=\left(x^{2}\right)^{2}, x^{8}=\left(x^{4}\right)^{2},\) and \(x^{3}=x\left(x^{2}\right) .\) $$ x^{4}+2 x^{2}+1=0 $$

Determine whether the points in each set are collinear. Explain how you know. $$ (-2,13),(1.5,-4.5),(3,-12) $$

In Chapter \(4,\) you solved inequalities involving linear relationships. For this problem, use the same ideas to solve inequalities involving quadratic relationships. a. First use the quadratic formula to solve \(x^{2}-3 x-7=0\) b. Use the information from Part a to help graph \(y=x^{2}-3 x-7\) You may want to plot some additional points. Use your solutions and graph to solve each inequality. c. \(x^{2}-3 x-7<0\) d. \(x^{2}-3 x \geq 7\) e. \(x^{2}-3 x \leq 7\)

Expand each expression. $$ 9 c(-8+7 c) $$

Solve each equation. $$ 3(x-3)^{2}=30 $$
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