Two numbers have a sum of 14. four times the larger number is 4 less than 6 times the smaller number. What are the numbers

Let's denote the two numbers as \( x \) and \( y \), where \( x \) is the larger number and \( y \) is the smaller number.

We are given two conditions:
1. The sum of the two numbers is 14: 
   \[
   x + y = 14 \tag{1}
   \]
2. Four times the larger number is 4 less than six times the smaller number:
   \[
   4x = 6y - 4 \tag{2}
   \]

### Step 1: Solve for one variable from Equation (1)
From equation (1), we can express \( x \) in terms of \( y \):
\[
x = 14 - y
\]

### Step 2: Substitute this expression into Equation (2)
Substitute \( x = 14 - y \) into equation (2):
\[
4(14 - y) = 6y - 4
\]

### Step 3: Simplify the equation
Expand the left-hand side:
\[
56 - 4y = 6y - 4
\]

### Step 4: Solve for \( y \)
Move all terms involving \( y \) to one side:
\[
56 + 4 = 6y + 4y
\]
\[
60 = 10y
\]
\[
y = 6
\]

### Step 5: Solve for \( x \)
Now that we know \( y = 6 \), substitute this back into equation (1) to find \( x \):
\[
x + 6 = 14
\]
\[
x = 8
\]

### Final Answer:
The two numbers are \( x = 8 \) and \( y = 6 \).

Would you like further details or explanations?

Here are 5 related questions:
1. How would the solution change if the sum of the two numbers were 20 instead of 14?
2. What if the condition was 4 times the smaller number is 6 less than 6 times the larger number?
3. Can you generalize this method for any sum and similar conditions?
4. What happens if we swap the terms for the larger and smaller number in the second condition?
5. How can this be represented graphically?

**Tip:** For word problems, always try to translate the problem into equations by identifying the relationships between the quantities.

Two numbers have a sum of 14. Four times the larger number is 4 less than 6 times the smaller number. What are the numbers?

Learn how to solve a system of linear equations to find two numbers that sum to 14 and satisfy a specific linear relationship. This step-by-step solution covers algebraic methods suitable for students in Grades 8-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation