/*! 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 41 Solve the system using any metho... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 41

Solve the system using any method. $$ \begin{array}{l} 4(x-2)=6 y+3 \\ \frac{1}{4} x-\frac{3}{8} y=-\frac{1}{2} \end{array} $$

Short Answer

Expert verified
The system has no solution.

Step by step solution

01

Simplify the first equation

Start by simplifying the first equation: \[4(x-2)=6y+3\] Distribute the 4: \[4x - 8 = 6y + 3\] Rearrange to put all terms on one side: \[4x - 6y = 11\]
02

Simplify the second equation

Simplify the second equation: \[\frac{1}{4}x - \frac{3}{8}y = -\frac{1}{2}\] To clear the fractions, multiply the entire equation by 8: \[2x - 3y = -4\]
03

Set up the system of equations

The simplified system is: \[\begin{cases} 4x - 6y = 11 \ 2x - 3y = -4 \end{cases}\]
04

Eliminate one variable

Let's eliminate variable \(y\). Multiply the second equation by 2 to align with the first equation: \[\begin{cases} 4x - 6y = 11 \ 4x - 6y = -8 \end{cases}\]
05

Subtract the equations

Subtract the second equation from the first: \[ (4x - 6y) - (4x - 6y) = 11 - (-8) \] Which simplifies to: \[0 = 19\] This is a contradiction, meaning the system has no solution.

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.

linear equations
Linear equations are mathematical expressions involving variables raised to the power of one. They take the general form of \(ax + by = c\), where \(a\), \(b\), and \(c\) are constants. These equations represent straight lines when graphed on a coordinate plane. Understanding how to handle these equations is crucial for solving various algebraic problems. The given system involves two linear equations which we need to simplify and manipulate to find the solution.
methods of elimination
One of the most powerful techniques to solve systems of linear equations is the method of elimination. This method involves combining the equations to eliminate one of the variables, making it easier to solve for the remaining variable. Here's how it works:
  • First, align the terms of both equations.
  • Then, multiply one or both equations to create coefficients that are opposites for one of the variables.
  • Next, add or subtract the equations to eliminate that variable.
  • Solve for the remaining variable, and substitute back to find the eliminated variable.
In our example, after simplifying the equations, we multiplied the second equation by 2 to align the coefficients of \(y\) with the first equation. This allowed us to eliminate \(y\) and attempt to find \(x\).
contradiction in algebra
In algebra, a contradiction is a situation where an equation or a system of equations generates an impossible result. For instance, if simplifying a system leads to a statement like \(0 = 19\), this is a contradiction. It indicates that there's no value of the variables that can satisfy both equations simultaneously. This was seen in our problem when we subtracted the second simplified equation from the first and ended up with the contradictory statement \(0 = 19\).
no solution system
A system of equations can have no solution when the equations represent parallel lines. Parallel lines never intersect, hence there are no common points (solutions) that satisfy both equations simultaneously. Systems with no solutions are known as inconsistent systems. In our exercise, the simplified equations ultimately led to a contradiction which indicated parallel lines. Therefore, the system has no solution as they don't intersect at any point on the graph.

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

Solve the system by using any method. If a system does not have one unique solution, state whether the system is inconsistent or whether the equations are dependent. (See Examples \(5-6\) ) $$ \begin{array}{l} 3 x-4 y=6 \\ 9 x=12 y+4 \end{array} $$

Solve the system of equations by using the addition method. (See Examples \(3-4)\) $$ \begin{array}{l} 5 x-2 y=-2 \\ 3 x+4 y=30 \end{array} $$

To protect soil from erosion, some farmers plant winter cover crops such as winter wheat and rye. In addition to conserving soil, cover crops often increase crop yields in the row crops that follow in spring and summer. Suppose that a farmer has 800 acres of land and plans to plant winter wheat and rye. The input cost for 1 acre for each crop is given in the table along with the cost for machinery and labor. The profit for 1 acre of each crop is given in the last column. $$ \begin{array}{|l|c|c|c|} \hline & \begin{array}{c} \text { Input Cost } \\ \text { per Acre } \end{array} & \begin{array}{c} \text { Labor/Machinery } \\ \text { Cost per Acre } \end{array} & \begin{array}{c} \text { Profit } \\ \text { per Acre } \end{array} \\ \hline \text { Wheat } & \$ 90 & \$ 50 & \$ 42 \\ \hline \text { Rye } & \$ 120 & \$ 40 & \$ 35 \\ \hline \end{array} $$ Suppose the farmer has budgeted a maximum of $$\$ 90,000$$ for input costs and a maximum of $$\$ 36,000$$ for labor and machinery. a. Determine the number of acres of each crop that the farmer should plant to maximize profit. (Assume that all crops will be sold.) b. What is the maximum profit? c. If the profit per acre for wheat were $$\$ 40$$ and the profit per acre for rye were $$\$ 45$$, how many acres of each crop should be planted to maximize profit?

A manufacturer produces two models of a gas grill. Grill A requires 1 hr for assembly and \(0.4 \mathrm{hr}\) for packaging. Grill \(B\) requires 1.2 hr for assembly and 0.6 hr for packaging. The production information and profit for each grill are given in the table. (See Example 4\()\) $$ \begin{array}{|l|c|c|c|} \hline & \text { Assembly } & \text { Packaging } & \text { Profit } \\ \hline \text { Grill A } & 1 \mathrm{hr} & 0.4 \mathrm{hr} & \$ 90 \\ \hline \text { Grill B } & 1.2 \mathrm{hr} & 0.6 \mathrm{hr} & \$ 120 \\ \hline \end{array} $$ The manufacturer has \(1200 \mathrm{hr}\) of labor available for assembly and \(540 \mathrm{hr}\) of labor available for packaging. a. Determine the number of grill A units and the number of grill B units that should be produced to maximize profit assuming that all grills will be sold. b. What is the maximum profit under these constraints? c. If the profit on grill A units is $$\$ 110$$ and the profit on grill \(\underline{B}\) units is unchanged, how many of each type of grill unit should the manufacturer produce to maximize profit?

Describe the solution set to the system of inequalities. \(x \geq 0, y \geq 0, x \leq 1, y \leq 1\)
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Decision Maths

Read Explanation

Discrete Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Logic and Functions

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.