/*! 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 90 In the following exercises, solv... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 90

In the following exercises, solve the systems of equations by substitution. $$ \left\\{\begin{array}{l} 4 x+y=10 \\ x-2 y=-20 \end{array}\right. $$

Short Answer

Expert verified
The solution to the system is \( x = 0 \) and \( y = 10 \).

Step by step solution

01

Solve one equation for one variable

First, solve the second equation for x: Given: ewline \[ x - 2y = -20 \] Add 2y to both sides: \[ x = 2y - 20 \]
02

Substitute into the other equation

Next, substitute \( x = 2y - 20 \) into the first equation: Given: ewline \[ 4x + y = 10 \] Substitute \( x = 2y - 20 \):\[ 4(2y - 20) + y = 10 \]
03

Simplify and solve for y

Simplify the equation: \[ 8y - 80 + y = 10 \] Combine like terms: \[ 9y - 80 = 10 \] Add 80 to both sides: \[ 9y = 90 \] Divide both sides by 9: \[ y = 10 \]
04

Solve for x

Substitute the value of y back into the equation \( x = 2y - 20 \): \[ x = 2(10) - 20 \] Simplify: \[ x = 20 - 20 \] \[ x = 0 \]
05

Verify the solution

Substitute \( x = 0 \) and \( y = 10 \) into the original equations to ensure they hold true: First equation: \[ 4(0) + 10 = 10 \] True. Second equation: \[ 0 - 2(10) = -20 \] True. Thus, the solution \( x = 0 \) and \( y = 10 \) is correct.

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 that represent straight lines when graphed. They typically take the form \ ax + by = c \, where \ a \ and \ b \ are coefficients, \ x \ and \ y \ are variables, and \ c \ is a constant. These equations can be part of a system of linear equations, like in the given problem. The goal here is to find values for \ x \ and \ y \ that satisfy both equations simultaneously.
In the given exercise, we work with two linear equations:
\[ 4x + y = 10 \]
and
\[ x - 2y = -20 \].

This system represents two lines on a graph, and solving it means finding the point where these two lines intersect.
Algebraic Substitution
Algebraic substitution is a useful method for solving systems of equations. The idea is to solve one equation for one of the variables and then substitute this expression into the other equation.
Here's a quick step-by-step guide:
  • Solve one of the equations for one variable.
  • Substitute that expression into the other equation.
  • Simplify and solve for the remaining variable.
In this problem, we first solve the second equation for \ x \:
\[ x - 2y = -20 \]
Adding \ 2y \ to both sides, we get:
\[ x = 2y - 20 \]
Then, we substitute this expression for \ x \ into the first equation:
\[ 4(2y - 20) + y = 10 \].

This allows us to solve for \ y \, simplifying our system to one equation with one unknown.
Variables and Constants
Understanding variables and constants is crucial in algebra. Variables are symbols that represent unknown values and can change, while constants are fixed values that do not change.
In our problem, \ x \ and \ y \ are variables, and the numbers 4, 1 (implicit in front of y), -20, and 10 are constants.
When we isolate \ y \ in our equations:
  • First, we rewrite \ x \ in terms of \ y \: \[ x = 2y - 20 \].
  • Then, we substitute this into the first equation, giving us constants \ 8 \ and -80 alongside variable \ y \: \[ 8y - 80 + y = 10 \].
Combining like terms and solving for \ y \ leads us to:
\[ y = 10 \].

With \ y \ known, substituting back to find \ x \ involves straightforward arithmetic with constants:
\[ x = 2(10) - 20 \], giving us \ x = 0 \.
Solution Verification
After solving for the variables, it's essential to verify that these solutions satisfy the original equations. This step ensures there's no mistake in the calculation process.
Given our solutions \ x = 0 \ and \ y = 10 \:
  • Substitute \ x = 0 \ and \ y = 10 \ into the first equation:
    \[ 4(0) + 10 = 10 \] which holds true.
  • Substitute the same values into the second equation:
    \[ 0 - 2(10) = -20 \] which also holds true.
These verifications confirm our solution: \ x = 0 \ and \ y = 10 \ is correct and satisfies both original equations. Always verify your solutions to confirm accuracy and correctness in solving systems of 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

Laurie was completing the treasurer's report for her son's Boy Scout troop at the end of the school year. She didn't remember how many boys had paid the \(\$ 15\) full-year registration fee and how many had paid the \(\$ 10\) partial-year fee. She knew that the number of boys who paid for a full-year was ten more than the number who paid for a partial-year. If \(\$ 250\) was collected for all the registrations, how many boys had paid the full-year fee and how many had paid the partial-year fee?

Caitlyn sells her drawings at the county fair. She wants to sell at least 60 drawings and has portraits and landscapes. She sells the portraits for \(\$ 15\) and the landscapes for \(\$ 10\). She needs to sell at least \(\$ 800\) worth of drawings in order to earn a profit. (a) Write a system of inequalities to model this situation. (b) Graph the system. (c) Will she make a profit if she sells 20 portraits and 35 landscapes? (d) Will she make a profit if she sells 50 portraits and 20 landscapes?

In the following exercises, solve each system by graphing. $$ \left\\{\begin{array}{l} y \geq \frac{3}{4} x-2 \\ y<2 \end{array}\right. $$

Sarah left Minneapolis heading east on the interstate at a speed of \(60 \mathrm{mph}\). Her sister followed her on the same route, leaving two hours later and driving at a rate of \(70 \mathrm{mph}\). How long will it take for Sarah's sister to catch up to Sarah?

In the following exercises, translate to a system of equations and solve. Two angles are supplementary. The measure of the larger angle is four more than three times the measure of the smaller angle. Find the measures of both angles.
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Mechanics Maths

Read Explanation

Probability and Statistics

Read Explanation

Logic and Functions

Read Explanation

Statistics

Read Explanation

Decision Maths

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.