Sure! Below is the structured response for your portfolio on solving linear equations, including answers to the questions.
Solving Linear Equations Portfolio Worksheet
Question 1: Determine the number of solutions for the following equations.
- \(2x + 3 = 2x + 5\) (square)
- \(y = 4x + 7\) (underline)
- \(3y - 6 = 3(y - 2)\) (circle)
- \(x - y = 3\) (underline)
- \(4x + 2y = 8\) (circle)
- \(x + 2 = x + 5\) (square)
Summary of Solutions:
-
Infinite Solutions:
- \(3y - 6 = 3(y - 2)\) (circle)
- \(4x + 2y = 8\) (circle)
-
No Solutions:
- \(2x + 3 = 2x + 5\) (square)
- \(x + 2 = x + 5\) (square)
-
One Solution:
- \(y = 4x + 7\) (underline)
- \(x - y = 3\) (underline)
Question 2: Infinite Solutions
Equations:
- \(3y - 6 = 3(y - 2)\)
- \(4x + 2y = 8\)
Solving the First Equation: \[ 3y - 6 = 3y - 6 \quad \text{(This simplifies to a true statement)} \]
Solving the Second Equation: \[ 4x + 2y = 8 \quad \Rightarrow \quad 2y = -4x + 8 \quad \Rightarrow \quad y = -2x + 4 \] This can be expressed in the standard form, revealing it represents the same line when rearranged.
Observation:
Both equations represent the same line, resulting in infinitely many solutions.
Interpretation of Infinite Solutions:
Having infinite solutions means every point on the line satisfies both equations.
Plugging Values into the Equations:
-
For \(x = 3\):
- \(4(3) + 2y = 8 \Rightarrow 12 + 2y = 8 \Rightarrow 2y = -4 \Rightarrow y = -2\)
-
For \(x = -2\):
- \(4(-2) + 2y = 8 \Rightarrow -8 + 2y = 8 \Rightarrow 2y = 16 \Rightarrow y = 8\)
Both solutions work in the context of infinite solutions.
Question 3: No Solutions
Equations:
- \(2x + 3 = 2x + 5\)
- \(x + 2 = x + 5\)
Solving the First Equation: \[ 2x + 3 - 2x = 5 \Rightarrow 3 = 5 \quad \text{(This is a contradiction)} \]
Solving the Second Equation: \[ x + 2 - x = 5 \Rightarrow 2 = 5 \quad \text{(This is also a contradiction)} \]
Observation:
Both equations are contradictions and therefore have no solutions.
Interpretation of No Solutions:
No solutions means the lines are parallel and do not intersect.
Plugging Values into the Equations:
-
For \(x = 2\):
- For the first equation: \(2(2) + 3 = 2(2) + 5 \Rightarrow 4 + 3 \neq 4 + 5\)
- For the second equation: \(2 + 2 \neq 2 + 5\)
-
For \(x = -3\):
- For the first equation: \(2(-3) + 3 \neq 2(-3) + 5\)
- For the second equation: \(-3 + 2 \neq -3 + 5\)
Both values show no solutions exist in both systems.
Question 4: One Solution
Equations:
- \(y = 4x + 7\)
- \(x - y = 3\)
Solving the First Equation: No need to solve; it's already in slope-intercept form.
Solving the Second Equation: \[ x - y = 3 \Rightarrow y = x - 3 \] The two lines intersect at one point.
Observation:
Each equation represents a different line that intersects at a single point, indicating one solution.
Interpretation of One Solution:
One solution means the equations intersect at exactly one point.
Plugging the Solution \(x=4\) to Verify:
- Substitute \(x=4\) into each equation:
- \(y = 4(4) + 7 = 16 + 7 = 23\) (not a valid solution)
- \(x - y = 3 \Rightarrow 4 - 23 \neq 3\)
This false solution confirms only one true intersection point exists.
Question 5: Writing Linear Equations
Infinite Solutions Example:
- Equation 1: \(2x - y = 4\)
- Equation 2: \(4x - 2y = 8\) (multiple of Eq. 1).
No Solutions Example:
- Equation 1: \(x + 2y = 4\)
- Equation 2: \(x + 2y = 6\) (parallel lines).
One Solution Example:
- Equation 1: \(y = 2x - 1\)
- Equation 2: \(y = -0.5x + 3\) (intersecting lines).
Real World Scenario:
A vehicle travels from two different routes, both described by linear equations. Depending on the slope and intercepts, one route may have multiple stop signs (infinite solutions), another may be closed off completely (no solution), while the third may offer a single point of interest (one solution).
Final Remarks:
This worksheet reflects the various outcomes for linear equations in terms of solutions. Understanding these concepts is crucial for both algebraic success and real-world problem-solving.
I hope this meets your requirements!
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