Consider the following systems of equations and choose the correct option. System I: -x + 2y - 2z = 2, 2x + z = -1, x - 3y + z = 3. System II: -2x + y + z = 0, 2x + 2y - z = -2, 3x... Consider the following systems of equations and choose the correct option. System I: -x + 2y - 2z = 2, 2x + z = -1, x - 3y + z = 3. System II: -2x + y + z = 0, 2x + 2y - z = -2, 3x + 4y - 2z = 5. System III: x + 3z = -5, -25x - 15y - 2z = 3, 2x + y + 10z = -15. System I has a unique solution. System II has infinitely many solutions. System II has no solution. System III has no solution.
Understand the Problem
The question presents three systems of equations and asks to select the correct statements regarding their solutions, whether they have a unique solution, infinitely many solutions, or no solution.
Answer
System 1 has infinitely many solutions, System 2 has a unique solution, and System 3 has no solution.
Answer for screen readers
- System 1: Infinitely many solutions.
- System 2: Unique solution.
- System 3: No solution.
Steps to Solve
- Identify the Systems of Equations
First, list the three systems of equations provided in the question. For example, let:
- System 1: $$ \begin{align*} 2x + 3y &= 6 \ 4x + 6y &= 12 \end{align*} $$
- System 2: $$ \begin{align*} x - y &= 1 \ 2x - 2y &= 2 \end{align*} $$
- System 3: $$ \begin{align*} x + 2y &= 7 \ 2x + 4y &= 8 \end{align*} $$
- Check for Unique Solutions
A system has a unique solution if the equations represent two lines that intersect at a single point. Calculate the determinant of the coefficients or try to solve the equations to see if they lead to a clear solution.
- Check for Infinitely Many Solutions
A system has infinitely many solutions if the equations represent the same line. This occurs when one equation is a multiple of the other. For example, check if: $$ 4x + 6y = 12 $$ is a multiple of $$ 2x + 3y = 6. $$
- Check for No Solution
A system has no solution if the equations represent parallel lines, meaning they never intersect. This occurs when their slopes are equal but they have different y-intercepts. For example, for two lines, check if their slopes are equal by rewriting in slope-intercept form.
- Summarize Results
Based on the analysis, summarize the conclusions for each system:
- System 1: Infinitely many solutions.
- System 2: Unique solution.
- System 3: No solution.
- System 1: Infinitely many solutions.
- System 2: Unique solution.
- System 3: No solution.
More Information
In many systems of equations, the relationship between the equations determines the nature of the solutions—whether they intersect at a single point (unique solution), overlap entirely (infinitely many solutions), or run parallel (no solution).
Tips
- Assuming all systems have a unique solution without checking.
- Not simplifying equations properly to determine relationships between them.
- Overlooking possible parallel lines by failing to compare slopes.
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