/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 55 Solve each equation. $$\frac{c... [FREE SOLUTION] | 91影视

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Chapter 6: Problem 55

Solve each equation. $$\frac{c-4}{4}=\frac{c+4}{8}$$

Short Answer

Expert verified
The solution to the equation is \(c = 12\).

Step by step solution

01

Clear the fractions through cross multiplication

To clear the fractions, we multiply each side of the equation by 8 and 4. Doing so gives us: \(8(c - 4) = 4(c + 4)\) which simplifies to \(8c - 32 = 4c + 16\).
02

Isolate 'c'

Next we combine like terms on each side of the equation, firstly by subtracting 4c from each side: \(8c - 4c = 32 + 16\). This simplifies to \(4c = 48\).
03

Solve for 'c'

Finally, we solve for 'c' by dividing each side of the equation by 4: \(c = 48 / 4\)

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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 like solving a puzzle. When you come across an equation, your goal is to find the value of the variable that makes the equation true.
In the given problem, the equation is \(\frac{c-4}{4}=\frac{c+4}{8}\). This is a type of equation where the variable \(c\) is hidden within fractions on both sides.
Our task is to uncover that variable using a variety of techniques. As you progress, remember that whatever you do to one side of the equation, you must do to the other to keep it balanced, much like a seesaw. This golden rule is key in maintaining the balance in equations.
Cross Multiplication
Cross multiplication is a handy technique, especially when you're dealing with equations involving fractions. It helps clear the fractions, making the equation simpler to solve.
Here's how it works: You multiply each numerator by the denominator on the opposite side of the equals sign. For our example:
  • Multiply \((c - 4)\) by 8.
  • Multiply \((c + 4)\) by 4.
This turns our original equation, \(\frac{c-4}{4}=\frac{c+4}{8}\), into \(8(c - 4) = 4(c + 4)\).
Now, you've transformed the equation into something cleaner without fractions. It's like opening a tightly sealed box 鈥� the contents are the same, but they're much easier to handle now.
Isolation of Variables
The isolation of variables is an essential step in equation solving. Once the fractions have been dealt with, it's time to focus solely on finding out what the variable represents.
For the equation \(8c - 32 = 4c + 16\), your aim is to get \(c\) all by itself on one side of the equation.
Here's how you achieve that:
  • Subtract \(4c\) from both sides to eliminate \(c\) from one side: \(8c - 4c = -32 + 16\).
  • This simplifies the equation to \(4c = 48\).
Now, the variable \(c\) is closer to being isolated, and you are a step nearer to finding its value.
Clearing Fractions
Clearing fractions is often the first crucial step to simplify an equation and make it more straightforward to solve. When fractions are involved, they can make calculations cumbersome. By eliminating them, we simplify our task.
  • In the equation \(\frac{c-4}{4}=\frac{c+4}{8}\), cross multiplication is used to clear fractions.
  • This results in \(8(c - 4) = 4(c + 4)\) with no fractions, making further calculations cleaner and easier.
After clearing fractions, you'll notice that your equation is much more manageable, just like untangling a knot makes the rope easier to handle.

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Most popular questions from this chapter

Write each expression in simplest form. If it is already in simplest form, so indicate. Assume that no denominators are \(0 .\) $$\frac{3 y+x y}{3 x+x y}$$

Refer to the tax deductions listed in the table. Give each ratio in simplest form. $$\begin{array}{lc}\text { Item } & \text { Amount } \\\\\hline \text { Medical } & \$ 995 \\\\\text { Real estate tax } & \$ 1,245 \\\\\text { Contributions } & \$ 1,680 \\\\\text { Mortgage interest } & \$ 4,580 \\\\\text { Union dues } & \$ 225\end{array}$$ Find the total amount of deductions.

Set up and solve a proportion. An airplane ascends 100 feet as it flies a horizontal distance of 1,000 feet. The two triangles in the illustration are similar. How much altitude will it gain as it flies a horizontal distance of 1 mile? (Hint: 5,280feet = 1 mile.)

Write each expression in simplest form. If it is already in simplest form, so indicate. Assume that no denominators are \(0 .\) $$\frac{48}{16 y}$$

Write each expression in simplest form. If it is already in simplest form, so indicate. Assume that no denominators are \(0 .\) $$\frac{3 x+3 y}{x^{2}+x y}$$
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