Analyzing solutions to linear systems algebraically | Algebra I | Khan Academy

TL;DR

When the two given equations have the same slope but different y-intercepts, the system has no solution.

Transcript

We're given a system of equations here, and we're told to solve for x and y. Now, the easiest thing to do here, since in both equations they're explicitly solved for y, is say, well, if y is equal to that, and y also has to equal this second equation, then why don't we just set them equal to each other? Or another way to think about it is, if y is ... Read More

Key Insights

• 😫 Systems of equations can be solved by setting the equations equal to each other or substituting one equation into another.
• 🥺 If the resulting equation leads to a contradiction, such as 100 = 120, the system has no solution.
• 🏙️ The absence of a solution occurs when the two equations have the same slope but different y-intercepts, preventing the lines from intersecting.
• 🏙️ Two equations with the same slope and y-intercept represent coincident lines, resulting in infinitely many solutions.
• 🏙️ In contrast, two equations with different slopes and y-intercepts represent parallel lines that never intersect.
• ❓ Understanding the graphical interpretation of equations can provide insights into the solutions of a system.
• ❓ Solving systems of equations is a fundamental concept in algebra.

Questions & Answers

Q: How can we solve a system of equations by setting them equal to each other?

In this case, setting the equations equal to each other allows us to eliminate the y variable and solve for x. However, if the resulting equation is contradictory, it means there is no solution to the system.

Q: What happens if the two equations in the system have different y-intercepts?

When the y-intercepts are different, the lines represented by the equations also have different points of intersection with the y-axis. Therefore, the lines will never intersect and the system has no solution.

Q: Can two equations with the same slope have a solution?

Yes, two equations with the same slope can have a solution if they also have the same y-intercept. In that case, the lines represented by the equations will be the same and intersect at infinite points.

Q: Is it possible for a system of equations to have more than one solution?

Yes, a system of equations can have more than one solution. This occurs when the two equations represent lines that are coincident, meaning they are the same line. In this case, any point on the line is a solution to the system.

Summary & Key Takeaways

• To solve for x and y in a system of equations, one option is to set the equations equal to each other or substitute one equation into another.

• If the resulting equation leads to a contradiction (such as 100 = 120), the system has no solution.

• This occurs when the two equations have the same slope but different y-intercepts, preventing them from intersecting.