/*! 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 4 Solve the system of equations. I... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 4

Solve the system of equations. If the system does not have one unique solution, state whether the system is inconsistent or the equations are dependent. $$ \begin{array}{l} y=2 x-5 \\ \frac{1}{5} x-\frac{1}{10} y=1 \end{array} $$

Short Answer

Expert verified
The system is inconsistent; there is no solution.

Step by step solution

01

Write the equations in standard form

The first equation is already simplified: \( y = 2x - 5 \). Rewrite the second equation to eliminate the fractions: \( \frac{1}{5}x - \frac{1}{10}y = 1 \). Multiply through by 10 to clear the fractions: \( 2x - y = 10 \).
02

Substitute the expression for y from Equation 1 into Equation 2

Equation 1 gives \( y = 2x - 5 \). Substitute \( 2x - 5 \) for \( y \) in Equation 2: \( 2x - (2x - 5) = 10 \).
03

Simplify and solve for x

Simplify the equation: \( 2x - 2x + 5 = 10 \). This simplifies to \( 5 = 10 \).
04

Analyze the result

The equation \( 5 = 10 \) is a contradiction, which means the system of equations has no solution. Therefore, the system is inconsistent.

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.

Standard Form Equations
Standard form equations are written as \( Ax + By = C \), where \( A \), \( B \), and \( C \) are integers.
In our problem, Equation 1, \( y = 2x - 5 \), can be rewritten in standard form as:
\( -2x + y = -5 \) or equivalently \( 2x - y = 5 \).
The second equation, \( \frac{1}{5}x - \frac{1}{10}y = 1 \), can be converted to standard form by eliminating fractions. Multiply every term by 10 to get:
\( 2x - y = 10 \).
Standard form makes systems easier to solve using substitution or elimination. This is because it clearly shows the coefficients of \( x \) and \( y \), making it easier to manipulate and solve the equations.

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

Use a graphing utility to approximate the solution(s) to the system of equations. Round the coordinates to 3 decimal places. $$ \begin{array}{l} y=-0.7 x+4 \\ y=\ln x \end{array} $$

Determine the number of solutions to the system of equations. $$ \begin{aligned} x^{2}-y^{2} &=0 \\ |x| &=|y| \end{aligned} $$

Solve the system using any method. $$ \begin{array}{l} 0.05 x+0.01 y=0.03 \\ x+\frac{y}{5}=\frac{3}{5} \end{array} $$

A system of equations is given in which each equation is written in slope- intercept form. Determine the number of solutions. If the system does not have one unique solution, state whether the system is inconsistent or whether the equations are dependent. $$ \begin{array}{l} y=8 x-\frac{1}{2} \\ y=8 x-\frac{1}{2} \end{array} $$

Use a graphing utility to graph the solution set to the system of inequalities. \(y<\frac{4}{x^{2}+1}\) \(y>\frac{-2}{x^{2}+0.5}\)
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Statistics

Read Explanation

Calculus

Read Explanation

Discrete Mathematics

Read Explanation

Geometry

Read Explanation

Probability and Statistics

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.