## Solving a System of Linear Equations in Two Variables### Problem Statement:Set up a system of linear equations in two variables and solve for the unknown quantities.**Given:**- One number is twelve more than four times another.- Their sum is 27.**Goal:** Find the numbers.### Variables:Let $ x $ represent the smaller number and $ y $ represent the larger number.### Steps:**Part 1 of 4:**The statement "One number is twelve more than four times another" can be written as:\[ y = 4x + 12 \]**Part 2 of 4:**The sum of the numbers is given as 27. This can be represented by the equation:\[ x + y = 27 \]**Part 3 of 4:**Substitute the expression for $ y $ from the first equation into the second equation:\[ x + (4x + 12) = 27 \]\[ 5x + 12 = 27 \]**Part 4 of 4:**Solve for $ x $:\[ 5x + 12 = 27 \]\[ 5x = 27 - 12 \]\[ 5x = 15 \]\[ x = 3 \]Now, substitute $ x = 3 $ back into the first equation to solve for $ y $:\[ y = 4(3) + 12 \]\[ y = 12 + 12 \]\[ y = 24 \]### Solution:The smaller number $ x $ is 3, and the larger number $ y $ is 24. ### Conclusion:We have set up and solved a system of linear equations, determining that the two numbers are 3 and 24.Feel free to practice similar problems and strengthen your understanding of solving systems of linear equations!

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Set up a system of linear equations in two variables and solve for the unknown quantities. One number is twelve more than four times another. Their sum is 27. Find the numbers. Let x represent the smaller number and y represent the larger number. Part: 0 / 4 Part 1 of 4 "One number is twelve more than four times another" can be written as:

Set up a system of linear equations in two variables and solve for the unknown quantities. One number is twelve more than four times another. Their sum is 27. Find the numbers. Let x represent the smaller number and y represent the larger number. Part: 0 / 4 Part 1 of 4 "One number is twelve more than four times another" can be written as:

Algebra and Trigonometry (6th Edition)

6th Edition

ISBN:9780134463216

Author:Robert F. Blitzer

Publisher:Robert F. Blitzer

ChapterP: Prerequisites: Fundamental Concepts Of Algebra

Section: Chapter Questions

Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...

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## Solving a System of Linear Equations in Two Variables

### Problem Statement:
Set up a system of linear equations in two variables and solve for the unknown quantities.

**Given:**
- One number is twelve more than four times another.
- Their sum is 27.

**Goal:** Find the numbers.

### Variables:
Let $ x $ represent the smaller number and $ y $ represent the larger number.

### Steps:

**Part 1 of 4:**
The statement "One number is twelve more than four times another" can be written as:
\[ y = 4x + 12 \]

**Part 2 of 4:**
The sum of the numbers is given as 27. This can be represented by the equation:
\[ x + y = 27 \]

**Part 3 of 4:**
Substitute the expression for $ y $ from the first equation into the second equation:
\[ x + (4x + 12) = 27 \]
\[ 5x + 12 = 27 \]

**Part 4 of 4:**
Solve for $ x $:
\[ 5x + 12 = 27 \]
\[ 5x = 27 - 12 \]
\[ 5x = 15 \]
\[ x = 3 \]

Now, substitute $ x = 3 $ back into the first equation to solve for $ y $:
\[ y = 4(3) + 12 \]
\[ y = 12 + 12 \]
\[ y = 24 \]

### Solution:
The smaller number $ x $ is 3, and the larger number $ y $ is 24.

### Conclusion:
We have set up and solved a system of linear equations, determining that the two numbers are 3 and 24.

Feel free to practice similar problems and strengthen your understanding of solving systems of linear equations!

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