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Chapter 5: Problem 10

Solve each system by substitution. $$\begin{aligned}&y=-3 x-20\\\&6 x+2 y=5\end{aligned}$$

Short Answer

Expert verified
The given system of linear equations is: \[\begin{cases} y=-3x-20\\ 6x+2y=5 \end{cases}\] Substituting \(y\) from the first equation into the second equation, we get: \[6x + 2(-3x - 20) = 5\] Simplifying the equation, we encounter a contradiction, as we find that \(-40 = 5\). Since this statement is not true, there is no solution for x and y that would satisfy both equations. This means that the given system has no solution, implying that the lines are parallel and do not intersect.

Step by step solution

01

Substitute y from the first equation into the second equation.

We are given that \(y=-3x-20\). Now, we will substitute this into the second equation: $$6x + 2(-3x - 20) = 5$$
02

Simplify the equation and solve for x

To make the equation easier to work with, we will simplify by distributing and combining like terms: $$6x - 6x - 40 = 5$$ Now, after combining the x terms, we get: $$-40 = 5$$ However, this statement is not true, and therefore, there is no value of x that can satisfy the equation.
03

Analyzing the contradiction

Since we have a contradiction in our equation, it means that there is no solution for x and y that would satisfy both equations. Therefore, we can conclude that the given system of linear equations has no solution. This implies that the lines represented by these equations are parallel and do not intersect.

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Key Concepts

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

Understanding Systems of Equations
In mathematics, a system of equations consists of two or more equations that share the same set of variables. Here, we are dealing with a system that involves two variables, usually denoted as \( x \) and \( y \).
  • The goal when solving a system is to find the specific values of these variables that satisfy every equation in the system simultaneously.
  • There are several methods to solve systems of equations, such as the substitution method, elimination method, and graphing.
The substitution method, which is used in the provided exercise, involves solving one equation for one variable and then substituting this expression into the other equation. Solving systems of equations is crucial because it can represent real-world scenarios. Understanding how equations can work together to find a solution gives insight into problem-solving generally.
Introduction to Linear Equations
Linear equations are algebraic equations where each term is either a constant or the product of a constant and a single variable. Generally, they appear in the form \( ax + by = c \).
  • In the system we are looking at, we have two linear equations: \( y = -3x - 20 \) and \( 6x + 2y = 5 \).
  • Linear equations graph as straight lines on a coordinate plane, and their solutions are essentially points where the lines intersect.
Understanding linear equations is vital for solving more complex algebraic problems. These equations are foundational in algebra and are used extensively in various fields, from physics to economics.
Exploring No Solution Scenarios
A system of equations is said to have no solution when there is no set of values that satisfies all equations simultaneously. Mathematically, this occurs when the algebraic manipulation leads to a false statement, like in our problem where stepping through the substitution method gave us \(-40 = 5\), which is clearly untrue.
A no solution scenario generally has certain characteristics:
  • The result of solving the equations algebraically leads to a contradiction.
  • This often means that the graphical representation of the two equations results in parallel lines.
Such cases provide essential understanding, illustrating that not all systems of equations will have solutions, emphasizing the importance of analyzing the relationships between equations.
Understanding Parallel Lines
Parallel lines are lines in a plane that never meet, regardless of how far they are extended. Each pair of parallel lines is equidistant everywhere, lying in the same plane with the same slope.
  • The concept of parallelism is crucial in determining the solution nature of systems of linear equations.
  • In our given system, the realization that there is no solution also informs us that the lines representing the equations are parallel. This is because despite different constants \( c \), the proportional change between \( x \) and \( y \) (the slope) remains the same.
Understanding parallel lines helps in visualizing and solving systems of equations, clarifying why some equations have no points of intersection 鈥� a key insight when graphing and analyzing such problems.

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Most popular questions from this chapter

Write an equation of the line perpendicular to the given line and containing the given point. Write the answer in slope intercept form or in standard form, as indicated. Write an equation of the line perpendicular to \(x=7\) containing \((6,0)\)

Write the standard form of the equation of the line given the following information. contains \((-2,2)\) and \((8,7)\)

Write a system of equations and solve. Six White Castle hamburgers and one small order of french fries contain 955 calories. Eight hamburgers and two orders of fries contain \(1350 .\) Determine how many calories are in a White Castle hamburger and in a small order of french fries. (Source: www.whitecastle com)

The best-selling paper towel brands in 2002 were Bounty, Brawny, and Scott. Together they accounted for \(59 \%\) of the market. Bounty's market share was \(25 \%\) more than Brawny's, and Scott's market share was \(2 \%\) Iess than Brawny's. What percentage of the market did each brand hold in \(2002 ?\) (Source: \(U S A\) Today. Oct. 23 \(2003, p .3 B\) from Information 91影视, lnc.)

Solve each systen \(\begin{aligned} x+2 y+3 z &=4 \\\\-3 x+y &=-7 \\ 4 y+3 z &=-10 \end{aligned}\)
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