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Chapter 8: Problem 34

Solve each system by substitution. Determine whether each system is independent, inconsistent, or dependent. $$\begin{aligned} &2 x+y=9\\\ &4 x+2 y=10 \end{aligned}$$

Short Answer

Expert verified
The system is inconsistent with no solution.

Step by step solution

01

Solve for one variable in one equation

Choose the first equation, \[ 2x + y = 9 \] and solve for y. \[ y = 9 - 2x \]
02

Substitute the expression into the second equation

Take the expression for y and substitute it into the second equation: \[ 4x + 2(9 - 2x) = 10 \] This simplifies to \[ 4x + 18 - 4x = 10 \]
03

Simplify the resulting equation

Simplify the equation from Step 2: \[ 18 = 10 \] This is a contradiction.
04

Determine the nature of the system

Since the simplification leads to a contradiction (18 鈮� 10), the system has no solution. Therefore, the system is inconsistent.

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Key Concepts

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

substitution method
The substitution method is an algebraic technique used to solve systems of equations. It involves expressing one variable in terms of another using one of the equations.
For instance, in our system:
  • First, solve for one variable from one equation. In our example, from '2x + y = 9', we solve for y: y = 9 - 2x mph.
  • Next, substitute this expression for the variable into the other equation. For example: 4x + 2(9 - 2x) = 10
  • Simplify and solve the new equation to find the value of one variable. Then, substitute back to find the other variable.
The substitution method is particularly useful when one of the equations is easily solvable for one variable, leading to straightforward substitution in the second equation. It's a reliable method for both linear and more complex equations but involves carefully maintaining and simplifying the resulting expressions.
inconsistent systems
Inconsistent systems of equations are systems that have no solutions. This means that the equations describe lines that never intersect. Here鈥檚 how you identify an inconsistent system:
When you use methods like substitution or elimination and arrive at a contradiction鈥攕uch as an equation like
  • 18 = 10, 鈥攖his indicates that the system is inconsistent. No value will satisfy both equations simultaneously.
In our example, after substitution and simplifying, we find the equation math>18 = 10, with no solutions. Therefore, the lines represented by these equations are parallel and never meet.
An easy way to check if a system might be inconsistent is by comparing the slopes of the equations: if they are equal but the y-intercepts are different, the system will be inconsistent.
linear equations
Linear equations form the foundation of algebra and deal with variables that are linear. A linear equation in two variables (
    x, y) can be represented as ax + by = c .
Linear equations graph as straight lines in a coordinate plane. The solution to a system of two linear equations corresponds to the point(s) where the lines intersect.
  • If two lines intersect at a single point, the system is independent and has one solution.
  • If the lines are parallel (having the same slope but different y-intercepts), the system is inconsistent and has no solution.
  • If the lines coincide (having the same slope and y-intercept), the system is dependent and has an infinite number of solutions.
In our example, substituting one equation into another led to a contradiction, confirming the lines are parallel and the system is inconsistent.

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Most popular questions from this chapter

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