- For system (i), the coefficient ratios \(\frac{5}{10}
eq \frac{4}{-2}\) confirm a single unique solution.
- For system (ii), the equations are mutually contradictory, meaning there is no solution.
- For system (iii), the first two equations \(x+3y=26\) and \(x+3y=11\) directly contradict each other, resulting in no solution.
- For system (iv), the ratios \(\frac{5}{4}
eq \frac{-2}{2}\) confirm a single unique solution.
- Final Answer: (i) Single solution, (ii) No solution, (iii) No solution, (iv) Single solution.
Explanation
- Data and Problem Analysis
To analyze the systems of equations, we first represent each system in standard linear form \(ax + by = c\). For systems with two equations, we can use the ratios of coefficients to determine the nature of the solutions:
- A system has a unique solution if the ratio of the \(x\)-coefficients is not equal to the ratio of the \(y\)-coefficients (\(a_1/a_2
eq b_1/b_2\)).
- A system has no solution if the ratio of the \(x\)-coefficients is equal to the ratio of the \(y\)-coefficients, but not equal to the ratio of the constants (\(a_1/a_2 = b_1/b_2
eq c_1/c_2\)).
- A system has infinitely many solutions if all ratios are equal (\(a_1/a_2 = b_1/b_2 = c_1/c_2\)).
For systems with more than two equations, the system has a unique solution if they all intersect at one point, no solution if there is any contradiction between equations, and infinite solutions if the equations are redundant and consistent.
- System (i) Analysis
(i) Given:
\[5x + 4y = 6\]
\[10x - 2y = 7\]
We compare the ratios of the coefficients:
- Ratio of \(x\)-coefficients: \(\frac{5}{10} = \frac{1}{2} = 0.5\)
- Ratio of \(y\)-coefficients: \(\frac{4}{-2} = -2\)
Since \(0.5
eq -2\), the lines are not parallel, which means the system has a single unique solution.
- System (ii) Analysis
(ii) Given:
\[3x + 4y = 17\]
\[10y - x + 5y = 24 \implies -x + 15y = 24\]
\[5x = 12 - y \implies 5x + y = 12\]
We solve the system step-by-step: From the first two equations, solving for \(x\) and \(y\) gives \(y = \frac{89}{49}\) and \(x = \frac{159}{49}\). Substituting these values into the third equation, \(5(\frac{159}{49}) + (\frac{89}{49}) = \frac{795 + 89}{49} = \frac{884}{49} \approx 18.04\). Since \(18.04
eq 12\), there is a contradiction. Therefore, the system has no solution.
- System (iii) Analysis
(iii) Given:
\[x + 3y = 26\]
\[2x + 6y = 22 \implies x + 3y = 11\]
\[8y = 3\]
\[4x + 8y = 15\]
Comparing the first two equations, \(x + 3y = 26\) and \(x + 3y = 11\), we see an immediate contradiction because the same expression \((x + 3y)\) cannot equal both 26 and 11. Therefore, the system has no solution.
- System (iv) Analysis
(iv) Given:
\[y = \frac{5}{2}x \implies 5x - 2y = 0\]
\[4x + 2y = 10\]
We compare the ratios of the coefficients:
- Ratio of \(x\)-coefficients: \(\frac{5}{4} = 1.25\)
- Ratio of \(y\)-coefficients: \(\frac{-2}{2} = -1\)
Since \(1.25
eq -1\), the slopes are different and the lines must intersect at a single point. Therefore, the system has a single unique solution.
- Summary of Findings
In conclusion:
(i) Single solution
(ii) No solution
(iii) No solution
(iv) Single solution
Examples
Systems of linear equations are used in urban planning to determine the intersection of traffic patterns or utility lines. By modeling these as equations, planners can verify if there is a single point where everything meets (unique solution), if certain roads never meet (no solution), or if two proposed plans represent the exact same route (infinite solutions), ensuring infrastructure is designed efficiently and without conflicts.