Table of contents

0. Review of Algebra4h 16m
1. Equations & Inequalities3h 18m
2. Graphs of Equations43m
3. Functions2h 17m
4. Polynomial Functions1h 44m
5. Rational Functions1h 23m
6. Exponential & Logarithmic Functions2h 28m
7. Systems of Equations & Matrices4h 6m
8. Conic Sections2h 23m
9. Sequences, Series, & Induction1h 19m
10. Combinatorics & Probability1h 45m

7. Systems of Equations & Matrices

Two Variable Systems of Linear Equations

College Algebra

7. Systems of Equations & Matrices

Two Variable Systems of Linear Equations

2:39 minutes

Problem 75b

Textbook Question

For what value(s) of k will the following system of linear equations have no solution? infinitely many solutions? x - 2y = 3 -2x + 4y = k

Verified step by step guidance

Identify the coefficients of the variables in both equations. For the first equation, x - 2y = 3, the coefficients are 1 for x and -2 for y. For the second equation, -2x + 4y = k, the coefficients are -2 for x and 4 for y.

Notice that the coefficients of x and y in the second equation are exactly -2 times the coefficients of x and y in the first equation. This suggests that the two equations are multiples of each other.

To determine the conditions for no solution or infinitely many solutions, compare the constants on the right-hand side of the equations. The first equation has a constant of 3, and the second equation has a constant of k.

If the second equation is a multiple of the first, then for the system to have infinitely many solutions, the constant k must also be a multiple of 3 by the same factor. Calculate this factor based on the coefficients of x and y.

If k does not satisfy the condition from the previous step, then the system will have no solution because the lines represented by the equations will be parallel and will not intersect.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

System of Linear Equations

A system of linear equations consists of two or more linear equations involving the same set of variables. The solutions to the system are the values of the variables that satisfy all equations simultaneously. The system can have one solution, no solution, or infinitely many solutions, depending on the relationships between the equations.

Conditions for No Solution

A system of linear equations has no solution when the lines represented by the equations are parallel, meaning they have the same slope but different y-intercepts. This occurs when the equations are inconsistent, indicating that there is no point at which the lines intersect.

Conditions for Infinitely Many Solutions

A system of linear equations has infinitely many solutions when the equations represent the same line, meaning they are dependent. This occurs when one equation can be derived from the other through multiplication or addition, resulting in identical slopes and y-intercepts.