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

TL;DR

Operations on a system are valid when they preserve equality and the x, y pairs that satisfy both equations. For 2x + y = 8 and x + y = 5, multiplying the first equation by −1 and then adding the equations eliminates y, producing −x = −3; this leads to x = 3 and y = 2. Read on to see why each transformation creates an equivalent statement.

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: What does it mean to solve a system of equations?

Solving a system means finding an x, y pair that satisfies both equations simultaneously. Although each equation can have many solutions by itself, the pair must meet both constraints.

Q: How do you solve 2x + y = 8 and x + y = 5?

Multiply 2x + y = 8 by −1 to obtain −2x − y = −8, then add x + y = 5. This gives −x = −3, so x = 3; substituting 3 into x + y = 5 gives y = 2.

Q: Why does multiplying both sides of an equation by the same number preserve its solutions?

Both sides represent equal quantities, so applying the same multiplication to each side maintains the equality. For example, multiplying 2x + y = 8 by −1 produces the equivalent equation −2x − y = −8.

Q: Why can two equations in a system be added together?

If the left side of each equation equals its right side, adding the two left sides and the two right sides preserves equality. In the example, x + y equals 5, so adding that equation is equivalent to adding 5 to both sides of −2x − y = −8.

Q: How does adding the equations eliminate y in this example?

Adding −2x − y = −8 and x + y = 5 makes −y and y cancel. The resulting equation is −x = −3, which contains only one unknown.

Q: Why must the same operation be performed on both sides of an equation?

Performing the same operation on both sides maintains the equality between them. Thus, dividing both sides of −x = −3 by −1 gives x = 3 without changing the solution.

Q: How is y found after solving for x?

Substitute x = 3 into either original equation. Using x + y = 5 gives 3 + y = 5, and subtracting 3 from both sides gives y = 2.

Q: What makes two systems or equations equivalent?

They are equivalent when the same x, y pairs satisfy both forms. In this example, the original system, its transformed equations, and the statement x = 3 and y = 2 describe the same solution.

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.