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Chapter 4: Problem 2

Solve the equations and inequalities. $$\frac{y}{5}=4$$

Short Answer

Expert verified
The solution is \( y = 20 \).

Step by step solution

01

Understand the Equation

The given equation is \( \frac{y}{5} = 4 \). It states that the variable \( y \) divided by 5 equals 4.
02

Isolate the Variable

To solve for \( y \), multiply both sides of the equation by 5 to cancel out the division by 5. This gives us: \( \frac{y}{5} \times 5 = 4 \times 5 \).
03

Simplify the Equation

By performing the multiplication, the equation simplifies to: \( y = 20 \).
04

Verify the Solution

Check the solution by substituting \( y = 20 \) back into the original equation: \( \frac{20}{5} = 4 \). Since this holds true, \( y = 20 \) is correct.

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Key Concepts

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

Linear Equations
Linear equations are mathematical statements that show the equality of two expressions with one or more variables. In this case, our equation is linear because it involves only the first power of the variable y.
Linear equations can look like:
\[ ax + b = c \]
Here, a, b, and c are constants, and x is the variable. The goal is usually to find the value of the variable that makes the equation true.
Our given equation, \( \frac{y}{5} = 4 \), is a simple linear equation. Let's take a closer look at how we solve it.
Isolating the Variable
To solve a linear equation, typically the main step involves isolating the variable. This means getting the variable alone on one side of the equation.
For our equation, \( \frac{y}{5} = 4 \), y is divided by 5. To isolate y, we need to do the opposite of the division, which is multiplication.
So, we multiply both sides of the equation by 5. This cancels out the division by 5 on the left-hand side:
\[ \frac{y}{5} \times 5 = 4 \times 5 \]
On simplifying, we find:
\[ y = 20 \]
Now y is isolated, and we have found our solution.
Verifying Solutions
After isolating the variable and solving the equation, it's always crucial to verify the solution. This ensures our answer is correct.
For our problem, we found y = 20. To verify, substitute 20 back into the original equation:
\[ \frac{20}{5} = 4 \]
When you simplify, you get:
\[ 4 = 4 \]
This confirms our solution is correct since the left side equals the right side.
Verifying helps catch any mistakes made during calculations, ensuring that the solution is accurate.

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Most popular questions from this chapter

Group the equivalent fractions together: (a) \(-\frac{3}{4}\) (b) \(\frac{-3}{-4}\) (c) \(\frac{3}{-4}\) (d) \(-\frac{-3}{-4}\) (e) \(\frac{-3}{4}\) (f) \(-\frac{-3}{4}\) (g) \(-\frac{3}{-4}\)

Use a calculator to do the computation. Round off to three decimal places where necessary. $$\frac{12}{7} \cdot \frac{5}{8}$$

In each of the following exercises, perform the indicated operations. Express your answer as a single fraction reduced to lowest terms. $$\frac{5}{3 x y}+\frac{1}{6 y^{2}}$$

Solve the equations and inequalities. $$\frac{y+2}{14}-\frac{4 y+1}{7}=1$$

Make up a verbal problem involving money that would give rise to the following equation (be sure to indicate what the variable would represent): $$25 x+10(x+3)=275$$
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