/*! 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 12 Solve each system by the elimina... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 12

Solve each system by the elimination method. Check each solution. $$ \begin{array}{l} 5 x=y+5 \\ -5 x+2 y=0 \end{array} $$

Short Answer

Expert verified
The solution is \(x = 2\) and \(y = 5\).

Step by step solution

01

Align both equations

First, rewrite the given system of equations to clearly see their structure: \[ \begin{array}{l} 5x - y = 5 \ -5x + 2y = 0 \end{array} \]
02

Add the equations to eliminate x

Add the two equations together to eliminate the variable \(x\): \[ (5x - y) + (-5x + 2y) = 5 + 0 \] Simplifying this gives: \[ y = 5 \]
03

Substitute y back into one of the original equations

Substitute \(y = 5\) back into the first original equation to solve for \(x\): \[ 5x = y + 5 \] Which becomes: \[ 5x = 5 + 5 \] Thus: \[ 5x = 10 \] Solving for \(x\) gives: \[ x = 2 \]
04

Verify the solution

Substitute \(x = 2\) and \(y = 5\) into the second original equation to verify the solution: \[ -5(2) + 2(5) = 0 \] Simplifying: \[ -10 + 10 = 0 \] This confirms the 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.

solving systems of equations
When you have more than one equation with multiple variables, you're dealing with a system of equations. The goal is to find the values of these variables that satisfy all the equations simultaneously. For instance, in our given example, we have two equations:
  • 5x = y + 5
  • -5x + 2y = 0
To solve such systems, there are multiple methods we can use, including substitution, elimination, and graphing. Each method has its unique steps but all aim to find where the equations intersect or, in other words, where the variable values meet the conditions set by each equation.
For our example, we used the elimination method, which proved efficient in eliminating one variable (x) by adding the equations together.
linear equations
A linear equation is an algebraic equation where each term is either a constant or the product of a constant and a single variable. In simpler terms, the equation forms a straight line when plotted on a graph. In our example, the equations we have are linear:
  • 5x = y + 5
  • -5x + 2y = 0
When rearranged, both equations can be visualized as straight lines. Equality in the context of linear equations means both expressions have the same value. When plotting, the solution to these equations would be the point where the two lines intersect. Linear equations are powerful in describing relationships and are foundational in algebra.
substitution method
The substitution method is another effective technique for solving systems of equations. It involves solving one of the equations for a particular variable and then substituting that expression into the other equation. Here's how it could work with our initial example:
  • First, solve one equation for one variable: 5x = y + 5, which can be rearranged to y = 5x - 5.
  • Next, substitute this expression for y into the other equation: -5x + 2(5x - 5) = 0.
  • Simplify to find x: -5x + 10x - 10 = 0, giving x = 2.
  • Finally, substitute x back into the initial rearranged equation to find y: y = 5(2) - 5, yielding y = 5.
This method also confirms our values of x = 2 and y = 5, proving that substitution is a versatile tool 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

In 2017 , the average Facebook user in the United States spent 693 more minutes per month using Facebook than the average Instagram user spent using Instagram. The average Instagram user spent 36 fewer minutes per month using Instagram than the average Snapchat user spent using Snapchat. The total amount of time that average users of each social network spent per month using their respective networks was 1347 min. How much time did the average user spend on each social network?

Tickets for a Harlem Globetrotters show cost \(\$ 30\) for upper level, \(\$ 51\) for center court, or \(\$ 76\) for floor seats. Nine times as many upper level tickets were sold as floor tickets, and the number of upper level tickets sold was 55 more than the sum of the number of center court tickets and floor tickets. Sales of all three kinds of tickets totaled \(\$ 95,215\). How many of each kind of ticket were sold?

Solve each system by the elimination method. Check each solution. $$ \begin{array}{l} 5 x+8 y=10 \\ 24 y=-15 x-10 \end{array} $$

Traveling for \(3 \mathrm{hr}\) into a steady headwind, a plane flies \(1650 \mathrm{mi}\). The pilot determines that flying with the same wind for \(2 \mathrm{hr}\), he could make a trip of \(1300 \mathrm{mi} .\) Find the rate of the plane and the wind speed.

Extend the method of this section to solve each system. Express the solution in the form \((x, y, z, w)\) $$ \begin{array}{l} 3 x+y-z+w=-3 \\ 2 x+4 y+z-w=-7 \\ -2 x+3 y-5 z+w=3 \\ 5 x+4 y-5 z+2 w=-7 \end{array} $$
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Mechanics Maths

Read Explanation

Calculus

Read Explanation

Probability and Statistics

Read Explanation

Statistics

Read Explanation

Applied Mathematics

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.