What Is the Solution to the System of Equations Below?
Understanding how to solve a system of equations is essential for mastering algebra, calculus, and many real‑world problems. In this guide, we walk through a specific system, explain the underlying concepts, and demonstrate multiple solving techniques so you can pick the one that best fits the situation.
Introduction
A system of equations consists of two or more equations that share the same variables. Solving the system means finding the values of the variables that satisfy all equations simultaneously. The system we’ll focus on is:
[ \begin{cases} 3x + 4y = 11 \ 2x - 5y = -1 \end{cases} ]
At first glance, it looks like a simple pair of linear equations, but the process of finding the unique solution (or determining that none or infinitely many exist) involves several important steps. Let’s break it down But it adds up..
Graphical Insight
Before diving into algebraic methods, it’s helpful to visualize the equations as lines on the Cartesian plane Not complicated — just consistent..
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First equation: (3x + 4y = 11).
Slope: (-\frac{3}{4}).
y‑intercept: (y = \frac{11}{4}) when (x = 0). -
Second equation: (2x - 5y = -1).
Slope: (\frac{2}{5}).
y‑intercept: (y = \frac{1}{5}) when (x = 0) Small thing, real impact..
Because the slopes are different, the two lines intersect at a single point—exactly one solution. The graphical view confirms that a unique solution exists and motivates algebraic verification Simple, but easy to overlook..
Algebraic Techniques
1. Substitution Method
The substitution method is straightforward when one equation can be solved easily for one variable.
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Solve the first equation for (x) (or the second for (y)).
[ 3x + 4y = 11 ;;\Rightarrow;; 3x = 11 - 4y ;;\Rightarrow;; x = \frac{11 - 4y}{3} ] -
Substitute this expression into the second equation.
[ 2\left(\frac{11 - 4y}{3}\right) - 5y = -1 ] -
Clear the fraction by multiplying both sides by 3.
[ 2(11 - 4y) - 15y = -3 ] -
Simplify:
[ 22 - 8y - 15y = -3 ;;\Rightarrow;; 22 - 23y = -3 ] -
Isolate (y):
[ -23y = -25 ;;\Rightarrow;; y = \frac{25}{23} ] -
Back‑substitute to find (x):
[ x = \frac{11 - 4\left(\frac{25}{23}\right)}{3} = \frac{11 - \frac{100}{23}}{3} = \frac{\frac{253 - 100}{23}}{3} = \frac{\frac{153}{23}}{3} = \frac{153}{69} = \frac{51}{23} ]
Solution: ((x, y) = \left(\frac{51}{23}, \frac{25}{23}\right)) Worth keeping that in mind..
2. Elimination (Addition/Subtraction) Method
Elimination eliminates one variable by adding or subtracting the equations after appropriate scaling.
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Align the equations:
[ \begin{aligned} 3x + 4y &= 11 \quad &(1)\ 2x - 5y &= -1 \quad &(2) \end{aligned} ] -
Choose a coefficient to eliminate.
Take this case: eliminate (x) by multiplying (1) by 2 and (2) by 3:[ \begin{aligned} 6x + 8y &= 22 \quad &(1')\ 6x - 15y &= -3 \quad &(2') \end{aligned} ]
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Subtract (2') from (1') to remove (x):
[ (6x + 8y) - (6x - 15y) = 22 - (-3) ;;\Rightarrow;; 23y = 25 ] -
Solve for (y):
[ y = \frac{25}{23} ] -
Substitute back into one original equation to find (x).
Using (1):
[ 3x + 4\left(\frac{25}{23}\right) = 11 ;;\Rightarrow;; 3x = 11 - \frac{100}{23} ;;\Rightarrow;; 3x = \frac{253 - 100}{23} ;;\Rightarrow;; 3x = \frac{153}{23} ;;\Rightarrow;; x = \frac{51}{23} ]
The elimination method confirms the same solution: ((x, y) = \left(\frac{51}{23}, \frac{25}{23}\right)).
3. Matrix (Determinant) Method
For those comfortable with linear algebra, the system can be expressed as (AX = B):
[ A = \begin{pmatrix} 3 & 4 \ 2 & -5 \end{pmatrix}, \quad X = \begin{pmatrix} x \ y \end{pmatrix}, \quad B = \begin{pmatrix} 11 \ -1 \end{pmatrix} ]
The inverse of (A) (if it exists) is:
[ A^{-1} = \frac{1}{\det(A)} \begin{pmatrix} -5 & -4 \ -2 & 3 \end{pmatrix}, \quad \det(A) = (3)(-5) - (4)(2) = -15 - 8 = -23 ]
Thus:
[ X = A^{-1}B = \frac{1}{-23} \begin{pmatrix} -5 & -4 \ -2 & 3 \end{pmatrix} \begin{pmatrix} 11 \ -1 \end{pmatrix} = \frac{1}{-23} \begin{pmatrix} (-5)(11) + (-4)(-1) \ (-2)(11) + 3(-1) \end{pmatrix} = \frac{1}{-23} \begin{pmatrix} -55 + 4 \ -22 - 3 \end{pmatrix} = \frac{1}{-23} \begin{pmatrix} -51 \ -25 \end{pmatrix} = \begin{pmatrix} \frac{51}{23} \ \frac{25}{23} \end{pmatrix} ]
Again, the same solution emerges.
Scientific Explanation of Uniqueness
Why is there exactly one solution? The key lies in the determinant of the coefficient matrix (A). A non‑zero determinant ((\det(A) \neq 0)) guarantees that:
- The matrix (A) is invertible.
- The system has a unique solution.
Here, (\det(A) = -23 \neq 0), confirming uniqueness. If the determinant were zero, the lines would either be parallel (no solution) or coincident (infinitely many solutions).
Frequently Asked Questions (FAQ)
| Question | Answer |
|---|---|
| **Can I use a calculator to solve this system?Still, ** | Parallel lines have the same slope but different intercepts, so no point satisfies both equations—no solution. Most graphing calculators have a solve or matrix function that will return the exact fractions. |
| What if the lines overlap? | Absolutely. Practically speaking, if one equation is already solved for a variable, substitution is convenient. In practice, |
| **What does it mean if the lines are parallel? Because of that, | |
| **What if the equations were non‑linear? | |
| **Is substitution always the best method?On the flip side, otherwise, elimination or matrix methods may be faster. ** | Not always. ** |
Conclusion
The system
[ \begin{cases} 3x + 4y = 11 \ 2x - 5y = -1 \end{cases} ]
has a unique solution:
[ \boxed{(x, y) = \left(\frac{51}{23}, \frac{25}{23}\right)} ]
Whether you choose substitution, elimination, or matrix algebra, each method consistently leads to the same result. Understanding the geometric interpretation, algebraic manipulation, and determinant criterion equips you to tackle any linear system confidently. Armed with these tools, you can now solve more complex systems, whether in pure mathematics, physics, economics, or engineering.
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Continuing smoothly from the conclusion, the unique solution derived for the system of equations (x = 2, y = 3, z = -1) serves as a cornerstone for more complex applications. In engineering design, such systems model interactions between multiple forces or parameters. Take this case: determining the precise tension required in three supporting cables of a suspension bridge to maintain equilibrium under specific load conditions relies on solving analogous multi-variable equations. The solution ensures structural integrity and safety by defining the exact state balance The details matter here..
Not obvious, but once you see it — you'll see it everywhere Simple, but easy to overlook..
What's more, this mathematical approach extends to economic modeling. Because of that, consider a scenario involving supply, demand, and production costs for three interconnected goods. Solving a system like this allows economists to predict market equilibrium points, where supply meets demand for each product simultaneously while accounting for cross-product influences. The unique solution pinpoints the optimal production levels and prices that clear all markets, demonstrating how abstract algebra translates into tangible policy and business strategies.
The computational methods used to solve these systems, such as matrix inversion or Gaussian elimination, are fundamental in computer-aided design (CAD) software and financial modeling tools. They enable engineers and analysts to rapidly simulate scenarios, optimize designs, and forecast outcomes with high precision. The ability to find a unique solution guarantees predictability in these models, crucial for reliable planning and decision-making.
At the end of the day, the solution to the given system of equations exemplifies the power of linear algebra in solving real-world problems with multiple interacting variables. From ensuring structural stability in engineering to optimizing complex economic systems, the unique solution provides a definitive answer where multiple factors must align perfectly. This mathematical rigor transforms abstract concepts into practical tools, driving innovation and efficiency across diverse fields, underscoring the indispensable role of precise problem-solving in advancing technology and understanding our complex world Simple, but easy to overlook..
