Reasoning with systems of equations | Equivalent systems of equations | Algebra I | Khan Academy

TL;DR

Learn the operations for manipulating and solving a system of equations by adding, subtracting, and multiplying equations.

Transcript

• [Presenter] So let's say I had the equation, two x plus y is equal to eight. This is an equation, one single equation with two unknowns and there's many different x, y pairs that would satisfy this equation. Now let's add a second equation, x plus y is equal to five. And once again, if we only looked at the second equation, there's many different... Read More

Key Insights

• ❓ A system of equations involves using multiple equations as constraints to find solutions.
• 👻 Manipulating equations by multiplying or adding allows for simplification and elimination of variables.
• 🙃 Multiplying both sides of an equation maintains equality and creates equivalent equations.
• 🪜 Adding equations together helps in eliminating variables and simplifying the system of equations.
• ❣️ Solving a system of equations involves finding x, y pairs that satisfy all equations simultaneously.
• 🙃 Any manipulation performed on one equation must be applied to both sides to maintain equality.
• 🙃 Adding a number to both sides of an equation does not change the solutions.

Questions & Answers

Q: How does multiplying both sides of an equation by the same number create an equivalent equation?

Multiplying both sides of an equation by the same number maintains equality because whatever operation is performed on one side must also be performed on the other side. This results in an equivalent equation with the same solutions.

Q: Why can we add two equations together when solving a system of equations?

Adding two equations together allows for the elimination of one variable, resulting in an equation with only one unknown. This is possible because when dealing with a system of equations, if two equations are equal, adding the two left sides and adding the two right sides will still be equal.

Q: How does adding a number to both sides of an equation help in solving a system of equations?

Adding a number to both sides of an equation maintains equality. When solving a system of equations, if we add the same number to both sides of an equation, the solutions remain unchanged. This is useful in simplifying the equations and eliminating variables.

Q: How do we find the values of x and y that satisfy both equations in a system?

After manipulating the equations to eliminate variables, we can solve for one variable. Once we have a value for one variable, we can substitute it into either equation to find the corresponding value for the other variable. This gives us the x and y values that satisfy both equations.

Summary & Key Takeaways

• A system of equations uses multiple equations as constraints to find values that satisfy all equations.

• Operations like multiplying both sides of an equation by the same number create equivalent equations that have the same solutions.

• Adding two equations together allows for the elimination of variables and helps in solving the system of equations.