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:

4x6y=10y+2(1)4x - 6y = 10y + 2 \quad \text{(1)} ty=12+2x(2)ty = \frac{1}{2} + 2x \quad \text{(2)}

We need to find the value of tt such that the system has no solution.

Step 1: Simplify both equations

Let's start with the first equation:

4x6y=10y+24x - 6y = 10y + 2

Rearrange to group the terms involving yy on one side and the terms involving xx on the other side:

4x6y10y=24x - 6y - 10y = 2 4x16y=24x - 16y = 2

Now, simplify the equation by dividing everything by 2:

2x8y=1(3)2x - 8y = 1 \quad \text{(3)}

Now, look at the second equation:

ty=12+2xty = \frac{1}{2} + 2x

Rearrange it to get:

ty2x=12(4)ty - 2x = \frac{1}{2} \quad \text{(4)}

Step 2: Analyze the system

We now have the system:

2x8y=1(3)2x - 8y = 1 \quad \text{(3)} ty2x=12(4)ty - 2x = \frac{1}{2} \quad \text{(4)}

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 y=mx+by = mx + b, where mm is the slope.

For equation (3):

2x8y=12x - 8y = 1

Solve for yy:

8y=2x+1-8y = -2x + 1 y = \frac{1}{4}x - \frac{1}{8} \quad \text{(slope is \frac{1}{4})}

For equation (4):

ty2x=12ty - 2x = \frac{1}{2}

Solve for yy:

ty=2x+12ty = 2x + \frac{1}{2} 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:

14=2t\frac{1}{4} = \frac{2}{t}

Step 5: Solve for tt

Solve the equation:

14=2t\frac{1}{4} = \frac{2}{t}

Multiply both sides by tt and 44:

t=8t = 8

Conclusion:

The value of tt that makes the system have no solution is 8\boxed{8}.


Would you like further details or clarification? Here are some related questions to explore:

  1. How would you solve a system of equations with two variables using substitution?
  2. What is the difference between dependent, independent, and inconsistent systems of equations?
  3. How do you identify parallel lines in a system of equations?
  4. How do you determine if a system of equations has infinitely many solutions?
  5. Can you graph the system of equations when t=8t = 8?

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