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Learn it your way
A short class with Sarah and her students. The other ways are under More.
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Today we're going to discuss linear equations in two variables. Can anyone tell me why we consider pairs of numbers for solutions?
Because there are two variables, we need two values that satisfy the equation?
Exactly! For example, in the equation 2x + 3y = 12, if I say x = 3, what could y be?
If x is 3, then y would be 2, right? Because 2(3) + 3(2) equals 12.
Spot on! So, (3, 2) is one solution. But there are infinitely many solutions as we can choose different values for x. What happens if x = 0?
Then y becomes 4!
Great! So (0, 4) is another solution. This shows that a linear equation in two variables has multiple pairs that satisfy its conditions.
Remember, each solution can be visualized as a point in the Cartesian plane.
Let's continue with our previous example. If I select x=1 for the equation 2x + 3y = 12, what do we get for y?
So we substitute and solve it: 2(1) + 3y = 12, which gives us y = rac{10}{3}.
Right! So (1, rac{10}{3}) is also a solution. Notice how we continuously generate new pairs? That's the beauty of linear equations.
Can we find solutions by throwing in negative values for x?
Absolutely! If x = -1, what’s y?
We substitute: 2(-1) + 3y = 12 so y = rac{14}{3}.
Perfect! Each of these solutions leads us to different points on the graph, illustrating the concept of infinite solutions visually.
Imagine you have a scenario where you need to budget money between two funds. If y is the amount in Fund B, and they need to equal a total of $500, how would we express that?
We could say x + y = 500, giving us another equation!
Exactly! Just like our previous equations, this also has infinite solutions. If x is $200, what’s y?
Then y would be $300, right?
Correct! Can anyone tell me why this helps in planning?
Since we have multiple solutions, we can allocate funds flexibly based on our priorities!
Exactly! Each point represents a possible budgeting strategy, allowing us to visualize options effectively.
Let's see if (2, 1) is a solution for 2x + 3y = 12. Who can check it?
Substituting x and y gives us 2(2) + 3(1) = 4 + 3 = 7, which is not equal to 12.
Great effort! So, (2, 1) does not satisfy the equation. How about (3, 2)?
Substituting shows we get 2(3) + 3(2) = 6 + 6 = 12! So, it works!
Correct! A solid way to validate solutions is by substitution. Always be sure to check the answers you've derived.
Remember, with infinite solutions, checking is essential to ensure you have the right pairs.
An equation of the first degree involving two variables.
A solution represented as (x, y) satisfying the equation.
The nature of linear equations in two variables leading to endless solutions.
A technique used to determine solutions by replacing a variable.
For every linear line you see, pairs of values must come free.
Alice and Bob share a total of 100 apples. For every apple Alice gets, Bob gets two. They can trade any way they like, but the total remains constant!
SPLIT: Substitute, Pair, Linear, Infinite, Test if valid.
SPL: Solutions, Pairs, Lines.
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What is a linear equation?
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What does an ordered pair (x, y) represent?
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What is substitution?
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An equation involving two variables that can be written in the form ax + by + c = 0.
A pair of numbers (x, y) that represents a solution to a linear equation.
A scenario where a linear equation in two variables can yield endless valid pairs (x, y).
A method of replacing a variable with a number or another expression in order to solve an equation.
This section explores the nature of solutions for linear equations in two variables, focusing on the infinite solutions that can be derived from such equations.
In this section, we carefully analyze the solutions of linear equations involving two variables, highlighting that each equation can yield infinitely many pairs fitting the equation. We explore various methods of finding solutions through examples and exercises, reinforcing the concept that each solution corresponds to a point on the Cartesian plane.
This section discusses the solutions to linear equations in two variables. Unlike linear equations in one variable that possess a unique solution, a linear equation in two variables provides us not just with one solution, but rather infinitely many pairs of values satisfying the equation form.
For example, the equation is examined, demonstrating how different pairs of sets, such as (3, 2) and (0, 4), fulfill the equation's requirements. Students learn to verify candidates' solutions through substitution and are encouraged to explore choosing values for one variable to find corresponding values for the other variable.
The section includes practical examples to derive multiple solutions from a single linear equation, explaining methods like setting one variable to zero to find intercepts on the Cartesian plane. Through engaging exercises and examples, learners understand that each valid solution corresponds to a point on the linear equation's graph, revealing the infinite nature of solutions in this context.
Example : Find four different solutions of the equation .
Solution: By inspection, is a solution because for
Now, let us choose . With this value of , the equation reduces to which has the unique solution so, is also a solution of the given equation.
Similarly, taking , the given equation reduces to . So, is a solution of . Finally, let us take . The given equation now reduces to , whose solution is . Therefore, is also a solution of the given equation. So four of the infinitely many solutions of the given equation are: