Solve The System Of Equations Using Substitution. 3x + 2y = 7 X = 3y + 6 (0, 2) (1, 2) (3, 1) (6, 0)
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Introduction
Solving systems of equations is a fundamental skill in algebra that allows us to find the point(s) where two or more equations intersect. One of the most effective methods for solving such systems, especially when one of the equations is already solved for a variable, is substitution. In this article, we will explore how to solve a system of equations using substitution, focusing on the specific system:
- 3x + 2y = 7
- x = 3y + 6
Additionally, we will evaluate the given points: (0, 2), (1, 2), (3, 1), and (6, 0) to determine which, if any, satisfy the system.
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Understanding the System of Equations
Before diving into the solution, it’s essential to understand the structure of the equations:
Equation 1: 3x + 2y = 7
This is a linear equation involving both variables x and y.
Equation 2: x = 3y + 6
This equation is already solved for x, making it ideal for substitution.
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What Is the Method of Substitution?
The substitution method involves solving one of the equations for one variable and then substituting that expression into the other equation. This process reduces the system to a single-variable equation, which can then be solved straightforwardly.
Steps in the substitution method:
- Solve one equation for one variable (here, equation 2 already expresses x in terms of y).
- Substitute this expression into the other equation.
- Solve for the remaining variable.
- Substitute back to find the other variable.
- Verify the solution by plugging the values into the original equations.
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Step-by-Step Solution
Let's apply the substitution method to our system:
Step 1: Identify the substitution
Equation 2 already gives x in terms of y:
- x = 3y + 6
Step 2: Substitute into the first equation
Replace x in the first equation with the expression from equation 2:
- 3(3y + 6) + 2y = 7
Step 3: Simplify and solve for y
Distribute:
- 9y + 18 + 2y = 7
Combine like terms:
- 11y + 18 = 7
Subtract 18 from both sides:
- 11y = 7 - 18
- 11y = -11
Divide both sides by 11:
- y = -1
Step 4: Find x using the value of y
Substitute y = -1 into equation 2:
- x = 3(-1) + 6
- x = -3 + 6
- x = 3
Solution: (x, y) = (3, -1)
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Verifying the Solution
It’s crucial to verify whether this solution satisfies both original equations.
- First equation: 3x + 2y = 7
Substituting x=3 and y=-1:
3(3) + 2(-1) = 9 - 2 = 7 ✔️
- Second equation: x = 3y + 6
Substituting y=-1:
x = 3(-1) + 6 = -3 + 6 = 3 ✔️
Both equations are satisfied, confirming that the solution is correct.
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Analyzing the Given Points
Now, let's evaluate the provided points to see which satisfy the system:
| Point | x | y | 3x + 2y | x (from point) | Is x = 3y + 6? | Satisfies the system? |
|---------|-----|-----|------------|--------------|-------------------|------------------------|
| (0, 2) | 0 | 2 | 3(0)+2(2)= 0+4=4 | 0 | 3(2)+6=6+6=12 | No, since 0 ≠ 12 |
| (1, 2) | 1 | 2 | 3(1)+2(2)= 3+4=7 | 1 | 3(2)+6=12 | No, since 1 ≠ 12 |
| (3, 1) | 3 | 1 | 3(3)+2(1)= 9+2=11 | 3 | 3(1)+6=3+6=9 | No, since 3 ≠ 9 |
| (6, 0) | 6 | 0 | 3(6)+2(0)= 18+0=18 | 6 | 3(0)+6=6 | No, since 6 ≠ 18 |
Conclusion: None of the provided points satisfy both equations simultaneously. The only solution to the system is (3, -1), which is not among the listed options.
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Graphical Interpretation
Visualizing the system can provide further understanding.
- The line 3x + 2y = 7 can be rewritten in slope-intercept form:
2y = -3x + 7
y = (-3/2)x + 7/2
- The line x = 3y + 6 can be rewritten as:
y = (x - 6)/3
Plotting these lines on a coordinate plane reveals their intersection point at (3, -1), confirming our algebraic solution.
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Practical Applications of Solving Systems Using Substitution
Understanding how to solve systems of equations using substitution has numerous real-world applications:
- Physics: Calculating the intersection point of two moving objects.
- Economics: Finding equilibrium points where supply and demand curves intersect.
- Engineering: Determining optimal points in design problems.
- Computer Science: Solving for variables in algorithms and models.
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Tips for Successful Substitution
- Always check if one of the equations is already solved for a variable, which simplifies substitution.
- Be careful with signs and arithmetic during substitution.
- Verify solutions by plugging values back into original equations.
- When dealing with complex systems, consider graphical or elimination methods as alternatives.
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Summary
In this article, we've demonstrated how to solve a system of equations using the substitution method. The key steps involved expressing x in terms of y, substituting into the other equation, solving for y, then finding x. The solution to the system:
- (x, y) = (3, -1)
was verified to satisfy both equations. We also examined the provided points and confirmed none satisfy the system, emphasizing the importance of algebraic verification.
Mastering substitution enhances problem-solving skills and provides a foundation for tackling more complex systems in mathematics and applied fields. Whether you're preparing for exams or working on real-world problems, understanding this method is a valuable tool in your mathematical toolkit.
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Frequently Asked Questions (FAQs)
Q1: Can the substitution method be used for all systems of equations?
A: It works best when one equation is already solved for a variable or can be easily rearranged. For more complex systems, other methods like elimination or graphing might be more efficient.
Q2: What if substitution leads to a contradiction?
A: If substitution results in a false statement (e.g., 0 = 5), the system has no solution—it's inconsistent.
Q3: How do I handle systems with more than two variables?
A: The substitution method can be extended to systems with more variables, but it often becomes more complex. Methods like matrix algebra or Gaussian elimination may be more suitable.
Q4: Is substitution always the best method?
A: Not necessarily. Sometimes, elimination or graphing can be quicker depending on the equations' form.
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Final Thoughts
Solving systems of equations is a foundational skill in algebra that opens doors to advanced mathematics and various scientific fields. The substitution method, in particular, offers an intuitive approach when one equation is already solved for a variable. Practice with different systems to become proficient, and always verify your solutions to ensure accuracy.
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