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Chapter 9: Problem 46

Solve the equation for \(\mathrm{x}.\) $$13=\frac{x}{5}$$

Short Answer

Expert verified
The solution to the equation is \(x = 65\).

Step by step solution

01

Multiplying Both Sides by 5

To solve for \(x\), we must remove the fraction by multiplying both sides of the equation by 5, as this is the denominator of the fraction. This will give us \(5 * 13 = x\).
02

Calculating the Value of x

Now, multiply 5 by 13 to find the value of \(x\). This will give us \(x = 65\).

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Key Concepts

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

Solving Equations
Solving linear equations is like solving a puzzle. The goal is to find the value of the unknown, often represented as \(x\). You do this by isolating \(x\) on one side of the equation. When dealing with equations such as \(13=\frac{x}{5}\), your aim is to eliminate any numbers or operations that are directly affecting \(x\). This often involves undoing operations, like division or multiplication, to make \(x\) stand alone.

Here's a simple step-by-step approach:
  • Identify the operation affecting the variable \(x\) (in this case, division by 5).
  • undo the operation by performing its inverse (multiply both sides by 5 in this equation).
  • Solve for \(x\) by simplifying both sides.
By consistently applying these steps, you can solve for \(x\) in many different types of equations.
Fractions in Equations
Fractions can make equations look tricky, but they are simply a number divided by another. In equations like \(13=\frac{x}{5}\), the fraction \(\frac{x}{5}\) represents \(x\) divided by 5. To make the equation easier to handle, you need to get rid of the fraction.

The most straightforward way to do this is to multiply every part of the equation by the denominator of the fraction:
  • Identify the denominator (in this example, it's 5).
  • Multiply both sides of the equation by this number to eliminate the fraction.
After multiplying through by 5, the fraction is gone, and the equation becomes \(13 \times 5 = x\).

Using this method allows you to turn a fractional equation into a simpler linear equation that is easier to solve.
Multiplication in Algebra
Multiplication is a fundamental operation in algebra used to manipulate equations and solve for unknowns. When solving the equation \(13=\frac{x}{5}\), multiplication helps to eliminate the fraction. Multiplying by 5 on both sides of the equation cancels out division by 5, simplifying the problem.

Remember these points about multiplication in algebra:
  • It is often used to clear fractions or distribute terms in equations.
  • Multiplying both sides of an equation by the same number keeps it balanced.
  • After multiplication, simplify the equation to find the solution for the unknown variable.
In this example, multiplying 13 by 5 gives \(65\), providing you with the solution \(x=65\). Understanding how to use multiplication effectively can make solving equations more straightforward and intuitive.

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Most popular questions from this chapter

MULTIPLE REPRESENTATIONSu are standing on a cliff above an ocean. You see a sailboat from your vantage point 30 feet above the ocean. a. Draw and label a diagram of the situation. b. Make a table showing the angle of depression and the length of your line of sight. Use the angles \(40^{\circ}\) , \(50^{\circ}, 60^{\circ}, 70^{\circ},\) and \(80^{\circ}\) . c. Graph the values you found in part (b), with the angle measures on the \(\mathrm{x}\) -axis. dredict the length of the line of sight when the angle of depression is \(30^{\circ}\) .

MATHEMATICAL CONNECTIONS\triangle EQU is equilateral and \(\Delta\) RGT is a right triangle with \(\mathrm{RG}=2,\) \(\mathrm{RT}=1,\) and \(\mathrm{m} \angle \mathrm{T}=90^{\circ},\) show that \(\sin \mathrm{E}=\cos \mathrm{G}\) .

CRITICAL THINKIN Explain why the area of \(\triangle \mathrm{ABCin}\) the diagram can be found using the formula Area \(=\frac{1}{2} a b \sin \mathrm{C}\) . Then calculate the area when \(\mathrm{a}=4, \mathrm{b}=7,\) and \(\mathrm{m\angleC}=40^{\circ}\) .

Solve the equation. $$\frac{12}{x}=\frac{3}{2}$$

WRITING Explain how you know the tangent ratio is constant for a given angle measure.
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