Solving a Pair of Linear Equations Methods Quiz

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Solving a Pair of Linear Equations

When dealing with two linear equations in two variables, x and y, we're often left with the task of determining the point(s) of intersection or the unique solution(s) to the system. In this guide, we'll examine four primary methods for solving a pair of linear equations: substitution, elimination method, graphical method, and word problems.

Substitution Method

This technique involves explicitly solving for one variable in one equation, substituting it into the other, and then solving for the remaining variable. For example, let's look at the system:

[ 2x + y = 5 ] [ 3x - y = 1 ]

Let's first solve for y in the first equation:

[ y = 5 - 2x ]

Then, substitute this expression for y in the second equation and solve for x:

[ 3x - (5 - 2x) = 1 ] [ 3x - 5 + 2x = 1 ] [ 5x - 5 = 1 ] [ 5x = 6 ] [ x = \frac{6}{5} ]

Now that we have x, we can find y using the original expression:

[ y = 5 - 2\cdot\frac{6}{5} ] [ y = \frac{5}{5} = 1 ]

Thus, the solution is ((\frac{6}{5}, 1)).

Elimination Method

In this strategy, we aim to eliminate one variable by adding or subtracting the equations in a specific manner. For example, let's look at a system with coefficients that are easy to multiply or divide:

[ 3x + 2y = 6 ] [ x - y = 1 ]

Let's first add the two equations, eliminating x:

[ (3x + 2y) + (x - y) = 6 + 1 ] [ 4y = 7 ] [ y = \frac{7}{4} ]

Now that we have y, find x using the second equation:

[ x - \frac{7}{4} = 1 ] [ x = 1 + \frac{7}{4} ] [ x = \frac{9}{4} ]

Thus, the solution is ((\frac{9}{4}, \frac{7}{4})).

Graphical Method

This technique involves plotting the equations in a coordinate plane and finding the point(s) of intersection. Let's revisit the first system:

[ 2x + y = 5 ] [ 3x - y = 1 ]

Plot the lines on a graph, and find the point(s) of intersection. The solution will be approximately ((\frac{6}{5}, 1)).

Word Problems

Solving word problems often requires us to translate the given information into a system of linear equations. For instance, consider the following problem:

Two trains depart simultaneously from point A, with train A moving at 50 mph and train B moving at 70 mph. After 2 hours, train A has traveled 240 miles, and train B has traveled 280 miles. To find out how far apart they are, we set up a system of linear equations:

[ x + 50t = 240 ] [ 70t + y = 280 ]

Solving this system, we find that the trains are approximately 180 miles apart.

Consistent and Inconsistent Systems

Solving a system of linear equations may result in either a unique solution, no solution (inconsistent), or infinitely many solutions (consistent with infinite solutions). The methods discussed above will help us determine which case we have for a given system. For example, if we have two lines that intersect at a single point, the system is consistent with a single solution. If the lines are parallel, the system is inconsistent with no solution. If the lines coincide, the system is consistent with infinitely many solutions.

Description

This quiz covers the various methods, including substitution, elimination, graphical, and word problems, for solving a pair of linear equations. Learn how to determine points of intersection, unique solutions, and how to identify consistent or inconsistent systems.

Solving a Pair of Linear Equations Methods Quiz

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Questions and Answers

What is the first step in solving a pair of linear equations using the substitution method?

Which method for solving a pair of linear equations involves eliminating one variable by adding or subtracting the equations?

If a pair of linear equations has no solution, it is referred to as a/an __ system.

What does the graphical method involve when solving a pair of linear equations?

What is the key concept in the elimination method for solving linear equations?

What is the solution to the system of equations: $3x + 2y = 6$ and $x - y = 1$?

When using the substitution method to solve linear equations, what should be done after solving for one variable?

When using the graphical method to solve the system of equations: $2x + y = 5$ and $3x - y = 1$, what is the approximate solution?

In the word problem provided where two trains are traveling with different speeds, how far apart are they after 2 hours?

When solving a system of linear equations that results in no solution, what can be said about the lines?

What happens when a system of linear equations has infinitely many solutions?

Which method involves adding or subtracting equations to eliminate one variable when solving a system of linear equations?

Study Notes