Answer :
Answer:
Hey! To study the effect of algebraic operations like addition and subtraction on linear equations in two variables graphically, you can follow these steps:
1. Start with two linear equations in the form:
- Equation 1: y = mx + c1
- Equation 2: y = nx + c2
2. Graph the two equations separately on the coordinate plane by plotting the y-intercepts (c1 and c2) and using the slopes (m and n) to determine additional points on the lines.
3. To study the effect of addition:
- Add the two equations together to form a new equation: y = (m + n)x + (c1 + c2).
- Graph the new equation on the same coordinate plane to observe the combined effect of the original equations.
4. To study the effect of subtraction:
- Subtract one equation from the other to form a new equation: y = (m - n)x + (c1 - c2).
- Graph the new equation on the same coordinate plane to observe how the subtraction affects the relationship between the original equations.
5. Analyze the graphs to observe how the addition and subtraction of the equations impact the intersection points, slopes, and overall relationship between the lines graphically.
By following these steps and visually representing the equations on a graph, you can observe the effects of algebraic operations like addition and subtraction on linear equations in two variables. Let me know if you need further clarification or assistance with any specific part of the process!
Answer:
Certainly! Let's explore the effect of algebraic operations (addition and subtraction) on linear equations in two variables graphically.
**Linear Equations in Two Variables:**
A linear equation in two variables can be represented as:
ax + by = c
where \(a\), \(b\), and \(c\) are constants, and \(x\) and \(y\) are the variables.
**Addition and Subtraction Phrases:**
When translating word problems into equations, we often encounter addition and subtraction phrases. Here are some examples:
1. **Addition Phrases:**
- "1 plus a" translates to \(1 + a\)
- "2 and b" translates to \(2 + b\)
- "4 more than c" translates to \(4 + c\)
2. **Subtraction Phrases:**
- "4 less d" translates to \(4 - d\)
- "g fewer than 7" translates to \(7 - g\)
- "6 less than a number" translates to \(x - 6\), where \(x\) represents the unknown number.
**Graphical Representation:**
To graphically represent linear equations, we plot the points that satisfy each equation. The point of intersection (if it exists) represents the solution to the system of equations.
For example, consider the system of equations:
1. \(2x + y = 15\)
2. \(3x - y = 5\)
The solution to this system is the ordered pair \((4, 7)\). We can verify this by substituting the values into both equations:
- For the first equation: \(2(4) + 7 = 15\) (True)
- For the second equation: \(3(4) - 7 = 5\) (True)
The intersection point \((4, 7)\) satisfies both equations simultaneously.
Step-by-step explanation: