Graph The Following System Of Equations.3x + Y = 63x + 3y = 12What Is The Solution To The System? There is a common question among students learning algebra and coordinate graphing techniques: how do we find the solution to a system of equations? Understanding how to graph and interpret the intersection point(s) of these equations is fundamental in algebra, as it helps to solve real-world problems involving multiple variables. This article will guide you through the process step-by-step, exploring the methods to graph the system, find the solution, and interpret the results effectively.
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Understanding Systems of Equations
Before diving into the graphing process, it’s important to understand what a system of equations is and what it represents.
What Is a System of Equations?
A system of equations consists of two or more equations with the same set of variables. The solutions to the system are the values of these variables that satisfy all equations simultaneously. In geometric terms, each equation represents a line (or curve), and the solution(s) are the point(s) where these lines intersect.Types of Solutions
- One solution: The lines intersect at exactly one point.
- No solution: The lines are parallel and do not intersect.
- Infinite solutions: The lines coincide, meaning they are the same line.
Analyzing the Given System of Equations
The system provided is:
- 3x + y = 6
- 3x + 3y = 12
Note: The original input "63x + Y = 63x + 3y = 12" appears to be a typo or formatting issue. Based on typical algebra problems, it’s likely intended as:
- First equation: 3x + y = 6
- Second equation: 3x + 3y = 12
Let’s proceed with this interpretation.
Step 1: Write Both Equations in Standard Form
- Equation 1: 3x + y = 6
- Equation 2: 3x + 3y = 12
Graphing the Equations
Graphing each line helps visualize the solution.
Step 1: Find the x- and y-intercepts
Intercepts are points where the line crosses the axes, making plotting easier.Equation 1: 3x + y = 6
- x-intercept: set y=0:
3x + 0 = 6 → x = 2
So, point: (2, 0)
- y-intercept: set x=0:
3(0) + y = 6 → y = 6
So, point: (0, 6)
Equation 2: 3x + 3y = 12
- x-intercept: set y=0:
3x + 0 = 12 → x = 4
So, point: (4, 0)
- y-intercept: set x=0:
3(0) + 3y = 12 → 3y=12 → y=4
So, point: (0, 4)
Step 2: Plot the Lines
Using the intercepts:- Line 1 passes through (2, 0) and (0, 6)
- Line 2 passes through (4, 0) and (0, 4)
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Finding the Solution Algebraically
While graphing provides a visual understanding, algebraic methods offer precise solutions.
Substitution Method
Since both equations are in standard form, substitution is straightforward.- From Equation 1: 3x + y = 6 → y = 6 - 3x
- Substitute y into Equation 2:
- Simplify:
- Combine like terms:
- Solve for x:
x = (-6)/(-6) = 1
- Find y using y = 6 - 3x:
y = 6 - 3(1) = 6 - 3 = 3
Solution: (x, y) = (1, 3)
Verification
Check the solution in both equations:- Equation 1: 3(1) + 3 = 3 + 3 = 6 ✔️
- Equation 2: 3(1) + 3(3) = 3 + 9 = 12 ✔️
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Interpreting the Graph and Solution
The intersection point (1, 3) confirms that the two lines cross at this coordinate, which is the unique solution to the system.
Understanding the Graph
- The lines intersect at exactly one point, indicating a consistent and independent system.
- The solution (1, 3) can be interpreted as the values of x and y that satisfy both equations simultaneously.
Real-World Applications
Such systems often appear in:- Business and economics (profit maximization, cost analysis)
- Physics (motion equations)
- Engineering (system design constraints)
Additional Methods for Solving Systems of Equations
While graphing and substitution are common, other methods include:
Elimination Method
- Multiply equations to align coefficients
- Subtract or add equations to eliminate a variable
- Solve for the remaining variable
Equation 1: 3x + y = 6
Equation 2: 3x + 3y = 12
Subtract Equation 1 from Equation 2:
(3x + 3y) - (3x + y) = 12 - 6
3x + 3y - 3x - y = 6
2y = 6 → y = 3
Plug back into Equation 1:
3x + 3 = 6 → 3x = 3 → x = 1
Solution remains (1, 3).
Graphing Method Summary
- Plot the equations on a coordinate plane
- Observe the intersection point
- Confirm the solution algebraically
Common Mistakes to Avoid
- Misreading equations: Ensure equations are correctly written and simplified.
- Incorrect plotting: Use accurate intercepts and scale.
- Calculation errors: Double-check arithmetic, especially when solving algebraically.
- Ignoring special cases: Parallel lines (no solution) or coincident lines (infinite solutions).
Conclusion
Graphing and solving systems of equations are essential skills in algebra that help understand relationships between variables. In our example, the system's solution at (1, 3) was found both graphically and algebraically, demonstrating the consistency of these methods. Whether for academic purposes or real-world applications, mastering these techniques allows you to analyze complex problems efficiently and accurately.
Remember, practice with different systems, including those with no solutions or infinite solutions, to strengthen your understanding of how lines behave in the coordinate plane. With patience and practice, you'll become proficient at graphing and solving systems of equations, making these mathematical tools invaluable for various disciplines.
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Keywords: graphing systems of equations, solving systems algebraically, intersection point, coordinate plane, x-intercept, y-intercept, substitution method, elimination method, algebra practice, math tutorial
