Math Problem Statement
4x-6y=10y+2 ty=1/2+2x
in the given system of equations, t is a. constant. if the system has no solution, what si the value of t
Solution
We are given the system of equations:
We need to find the value of such that the system has no solution.
Step 1: Simplify both equations
Let's start with the first equation:
Rearrange to group the terms involving on one side and the terms involving on the other side:
Now, simplify the equation by dividing everything by 2:
Now, look at the second equation:
Rearrange it to get:
Step 2: Analyze the system
We now have the system:
The system will have no solution if the lines represented by the two equations are parallel, meaning their slopes are equal but they have different intercepts.
Step 3: Express both equations in slope-intercept form
We will rewrite both equations in the form , where is the slope.
For equation (3):
Solve for :
y = \frac{1}{4}x - \frac{1}{8} \quad \text{(slope is \frac{1}{4})}
For equation (4):
Solve for :
y = \frac{2}{t}x + \frac{1}{2t} \quad \text{(slope is \frac{2}{t})}
Step 4: Set the slopes equal for parallel lines
For the system to have no solution, the slopes of the two lines must be equal. Thus:
Step 5: Solve for
Solve the equation:
Multiply both sides by and :
Conclusion:
The value of that makes the system have no solution is .
Would you like further details or clarification? Here are some related questions to explore:
- How would you solve a system of equations with two variables using substitution?
- What is the difference between dependent, independent, and inconsistent systems of equations?
- How do you identify parallel lines in a system of equations?
- How do you determine if a system of equations has infinitely many solutions?
- Can you graph the system of equations when ?
Tip: When solving for no solution or infinite solutions, focus on the slopes and intercepts of the lines.
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Math Problem Analysis
Mathematical Concepts
Linear Equations
Slope-Intercept Form
Systems of Equations
Parallel Lines
Formulas
Slope formula: y = mx + b
Solving for t in linear equations
Theorems
For no solution: slopes of lines must be equal but intercepts different
Suitable Grade Level
Grades 9-11
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