/*! 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 38 Solve the system using any metho... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 38

Solve the system using any method. $$ \begin{array}{l} 3(2 x-y)=2-x \\ x+\frac{5}{4} y=\frac{3}{2} \end{array} $$

Short Answer

Expert verified
The solution is \( x = \frac{28}{47} \) and \( y = \frac{34}{47} \).

Step by step solution

01

Simplify the equations

First, simplify each equation by eliminating parentheses and combining like terms. For the first equation, distribute the 3: \[ 3(2x - y) = 2 - x \] \[ 6x - 3y = 2 - x \] Rearrange it to standard form: \[ 6x + x - 3y = 2 \] \[ 7x - 3y = 2 \] The first equation is now: \[ 7x - 3y = 2 \] For the second equation: \[ x + \frac{5}{4} y = \frac{3}{2} \] The second equation is already in a simplified form.
02

Align the equations

Write both equations in a clear, aligned format for easy reference: \[ 7x - 3y = 2 \] \[ x + \frac{5}{4} y = \frac{3}{2} \]
03

Express one variable in terms of the other

Solve the second equation for \( x \): \[ x = \frac{3}{2} - \frac{5}{4} y \] Now, substitute this expression for \( x \) in the first equation.
04

Substitute and solve for y

Substitute \( x = \frac{3}{2} - \frac{5}{4} y \) into the first equation: \[ 7 \left( \frac{3}{2} - \frac{5}{4} y \right) - 3y = 2 \] Distribute and combine like terms: \[ 7 \times \frac{3}{2} - 7 \times \frac{5}{4} y - 3y = 2 \] \[ \frac{21}{2} - \frac{35}{4} y - 3y = 2 \] \[ \frac{21}{2} - \frac{35}{4} y - \frac{12}{4} y = 2 \] \[ \frac{21}{2} - \frac{47}{4} y = 2 \] Convert \( \frac{21}{2} \) to \( \frac{42}{4} \): \[ \frac{42}{4} - \frac{47}{4} y = 2 \] Combine terms and solve for \( y \): \[ \frac{42}{4} - 2 = \frac{47}{4} y \] \[ \frac{42}{4} - \frac{8}{4} = \frac{47}{4} y \] \[ \frac{34}{4} = \frac{47}{4} y \] Simplify: \[ 34 = 47y \] \[ y = \frac{34}{47} \]
05

Substitute y back to find x

Substitute \( y = \frac{34}{47} \) back into the expression for \( x \): \[ x = \frac{3}{2} - \frac{5}{4} \times \frac{34}{47} \] Calculate: \[ x = \frac{3}{2} - \frac{170}{188} \] Simplify \( \frac{170}{188} \): \[ x = \frac{3}{2} - \frac{85}{94} \] Convert \( \frac{3}{2} \) to \( \frac{141}{94} \): \[ x = \frac{141}{94} - \frac{85}{94} \] \[ x = \frac{56}{94} \] Simplify \( \frac{56}{94} \): \[ x = \frac{28}{47} \]

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

substitution method
The substitution method is a way to solve systems of linear equations by solving one equation for one variable, then substituting that solution into the other equation. This transforms the system into a single equation with one unknown. Here's how it works:
  • Solve one of the equations for one of the variables (e.g., solve for \(x\) in terms of \(y\)).
  • Substitute the resulting expression into the other equation.
  • Solve the new, single-variable equation for the unknown variable.
  • Substitute this solution back into one of the original equations to find the value of the other variable.
This method helps to eliminate one variable and makes solving the equations straightforward.
combining like terms
Combining like terms is a fundamental algebraic process used in solving equations. By simplifying expressions, you can make the equations more manageable. Here's what you need to do:
  • Identify terms with the same variable and exponent.
  • Add or subtract these terms to consolidate them into one term.
For example, in the problem we've solved:
  • We started with \(7x - 3y = 2\).
  • During substitution, we combined like terms involving \(y\).
  • We converted terms to have common denominators, then simplified.
It's important to keep equations tidy and in the simplest form possible for ease of calculation and clarity.
linear equations
Linear equations are equations that graph as straight lines. Each term in a linear equation has variables raised to the power of one. They take the form:
  • \(ax + by = c\)
In our example, we had two linear equations:
  • \(7x - 3y = 2\)
  • \(x + \frac{5}{4}y = \frac{3}{2}\)
The beauty of linear equations is their simplicity. You can solve them using various methods such as substitution, elimination, or graphical methods. Linear equations form the basis for more complex algebra and are crucial for understanding the principles of algebraic problem-solving.
standard form
The standard form of a linear equation is \(Ax + By = C\), where \(A\), \(B\), and \(C\) are constants. Here's why it's useful:
  • It makes equations easier to align and compare.
  • It provides a consistent format for solving systems of equations.
In our case, the problem required adjusting one equation to standard form:
  • We started with \(6x - 3y = 2 - x\).
  • Rearranged to get \(7x - 3y = 2\).
Using standard form simplifies the process of solving systems of equations. It standardizes methods like substitution and elimination, ensuring consistency and reducing errors.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Juan borrows \(\$ 100,000\) to pay for medical school. He borrows part of the money from the school whereby he will pay \(4.5 \%\) simple interest. He borrows the rest of the money through a government loan that will charge him \(6 \%\) interest. In both cases, he is not required to pay off the principal or interest during his 4 yr of medical school. However, at the end of \(4 \mathrm{yr}\), he will owe a total of \(\$ 19,200\) for the interest from both loans. How much did he borrow from each source?

Two particles begin at the same point and move at different speeds along a circular path of circumference \(280 \mathrm{ft}\). Moving in opposite directions, they pass in \(10 \mathrm{sec} .\) Moving in the same direction, they pass in \(70 \mathrm{sec} .\) Find the speed of each particle.

Solve the system. $$ \begin{array}{l} (x-1)^{2}+(y+1)^{2}=5 \\ x^{2}+(y+4)^{2}=29 \end{array} $$

Solve the system using any method. $$ \begin{array}{l} \frac{x-2}{8}+\frac{y+1}{2}=-6 \\ \frac{x-2}{2}-\frac{y+1}{4}=12 \end{array} $$

Determine if the ordered pair is a solution to the system of equations. (See Example 1\()\) \(y=-\frac{1}{5} x+2\) \(2 x+10 y=10\) a. (5,1) b. (-10,4)
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Pure Maths

Read Explanation

Mechanics Maths

Read Explanation

Probability and Statistics

Read Explanation

Geometry

Read Explanation

Logic and Functions

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.