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Today, we will discuss linear equations in two variables. Can anyone tell me what a linear equation looks like?
Isn’t it something like ax + by = c?
Exactly! And this equation represents a straight line on a graph. Who can explain why these equations can have multiple solutions?
Because the line intersects the x and y axes at different points, so there are lots of coordinates that satisfy the equation!
Great insight! This is foundational for understanding systems of equations.
To remember what a linear equation in two variables looks like, you can think of the acronym 'LIT' for 'Linear in Two dimensions.'
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Now let's dive into methods for solving these equations. What do you think the first method might be?
Maybe graphing them?
Correct! The graphical method involves plotting both equations and finding where they intersect. What happens at the intersection?
That point is the solution to the system of equations!
Yes! Now, let's discuss substitution next. Can anyone describe this method?
You solve one equation for one variable and use that in the other equation?
Exactly! And remember, for elimination, you either add or subtract the equations to eliminate one variable. To remember these methods, think 'GSE,' which stands for Graph, Substitute, Eliminate.
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Let’s take an example now. We have the equations \( x + y = 10 \) and \( x - y = 4 \). Who would like to solve this using the elimination method?
We can add both equations to eliminate \( y \). So, \( x + y + x - y = 10 + 4 \)?
Perfect! What do we get after simplifying?
That gives us \( 2x = 14 \), so \( x = 7 \).
Correct! Now substitute \( x \) back into the first equation to find \( y \). What do we get?
Substituting, we get \( 7 + y = 10 \), which means \( y = 3 \).
Fantastic! The solution is \( (7,3) \). So, to recap, we used elimination to find our solution. Remember, 'Two methods lead to one answer!'
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In this section, we explore how to solve linear equations in two variables, which can represent an infinite number of solutions. We discuss methods such as graphical, substitution, and elimination methods, providing examples to illustrate these concepts.
In this section, we learn how to solve linear equations containing two variables, such as those in the form of \( ax + by = c \). Unlike single-variable equations, these equations can have infinitely many solutions depicted graphically as straight lines on a coordinate plane.
To demonstrate the elimination method, consider the equations:
- \( x + y = 10 \)
- \( x - y = 4 \)
Using elimination, we add both equations to isolate one variable. This process yields \( x = 7 \) and substituting back gives \( y = 3 \). Thus, the solution is \( (7,3) \).
Understanding how to solve linear equations in two variables is vital for tackling more complex algebraic problems and applying these skills to various real-life situations.
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A linear equation in two variables has infinitely many solutions that can be represented as a straight line on a graph.
A linear equation in two variables involves two unknowns, typically denoted as x and y. The reason it has infinitely many solutions is that any point (x, y) that satisfies the equation can be plotted on a graph, forming a straight line. This means for any value of x, you can find a corresponding value of y, and vice versa. Thus, the equation expresses a relationship between the two variables and their values can vary widely while still fitting the equation.
Think of this like planning a road trip. If you know your distance and speed, you can calculate the corresponding time it will take. For example, if you travel at 60 miles per hour, for every hour spent driving (x), the distance covered (y) will change—60 miles in one hour, 120 miles in two hours, and so on, creating a straight line when plotted.
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🔸 Methods to Solve Systems of Equations
1. Graphical Method – Plot both equations and find the intersection.
2. Substitution Method – Solve one equation for one variable and substitute into the other.
3. Elimination Method – Add or subtract equations to eliminate one variable.
There are three main methods to solve systems of linear equations: 1) The Graphical Method involves plotting both equations on a graph and discovering where they intersect, which represents the solution. 2) The Substitution Method entails solving one of the equations for one variable and substituting its value into the other equation to find a solution. 3) The Elimination Method involves adding or subtracting both equations in order to eliminate one variable, making it easier to find the other.
Imagine two people walking towards each other on a street. The point where they meet represents the solution to the equations of their paths. Using different strategies (graphing their paths, finding how far one has walked to replace the other’s distance, or calculating how much closer they are with each step), you can determine that meeting point.
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🔹 Example
Solve:
• 𝑥 +𝑦 = 10
• 𝑥 −𝑦 = 4
Solution (Elimination method):
Add both equations:
2𝑥 = 14 → 𝑥 = 7
Substitute into first:
7+𝑦 = 10 → 𝑦 = 3
So, solution: (7,3)
In this example, we have two equations: x + y = 10 and x - y = 4. To solve using the Elimination Method, we add the two equations together to eliminate y, which simplifies to 2x = 14, allowing us to find x = 7. We then substitute this value back into one of the original equations (x + y = 10) to find y, giving us 7 + y = 10, hence y = 3. This means the solution (7, 3) represents a point where these two equations are satisfied.
Consider a scenario where you have $10 to spend on snacks. One type of snack costs a certain amount and the other type costs a different amount. If you have a budget and you want to buy a combination of snacks, figuring out how many of each you can buy is much like solving these equations. By calculating and finding the right combination that fits within your budget, you discover that you can purchase 7 of one snack and 3 of another.
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Key Concepts
Linear Equation: A relationship that can be expressed as ax + by = c.
Graphical Method: Finding solutions by graphing and looking for intersection points.
Substitution Method: Solving one equation for a variable and substituting into the other.
Elimination Method: Adding or subtracting equations to eliminate a variable.
See how the concepts apply in real-world scenarios to understand their practical implications.
Example of elimination method: For the equations x + y = 10 and x - y = 4, the elimination method gives the solution (7,3).
Graphing example: Plotting the equations x + y = 10 and y = 2x - 4 helps identify the intersection that represents the solution.
Use mnemonics, acronyms, or visual cues to help remember key information more easily.
In equations, we solve for x and y, find their point where they meet; oh my!
Once upon a time, in maths land, x and y were best friends. They crossed paths on a graph and found their meeting point, a magical spot that told their story.
Remember 'GSE': Graph, Substitute, Eliminate for solving systems of equations!
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Review the Definitions for terms.
Term: Linear Equation
Definition:
An algebraic equation that forms a straight line when graphed, typically expressed as ax + by = c.
Term: Graphical Method
Definition:
A technique for solving equations by plotting them on a graph to find their intersection.
Term: Substitution Method
Definition:
A method of solving systems of equations by replacing one variable with its equivalent expression from another equation.
Term: Elimination Method
Definition:
A strategy for solving systems of equations by adding or subtracting equations to eliminate one variable.
Term: Intersection Point
Definition:
The point where two lines intersect, representing the solution of a system of equations.