Answer :
To determine the nature of the solutions to the given linear equation [tex]\(6x - 2y = 10\)[/tex], we need to carefully analyze it step-by-step.
### Step 1: Identify the Type of Equation
The given equation [tex]\(6x - 2y = 10\)[/tex] is a linear equation in two variables (x and y). A linear equation in two variables typically represents a straight line in a two-dimensional coordinate system.
### Step 2: Analyze the Structure of the Equation
A linear equation in the form [tex]\(Ax + By = C\)[/tex] can have different types of solutions based on its structure:
1. Infinitely Many Solutions: This occurs when the same line is represented in different forms (i.e., the same equation can be rewritten in multiple ways).
2. 1 Solution: Typically, this applies in the context of systems of equations where two lines intersect at exactly one point.
3. No Solution: This happens if two lines are parallel and distinct, meaning they never intersect.
### Step 3: Solve for One Variable
To understand the solutions better, we can express [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex]:
[tex]\[ 6x - 2y = 10 \][/tex]
Solve for [tex]\( y \)[/tex] by subtracting [tex]\( 6x \)[/tex] from both sides:
[tex]\[ -2y = -6x + 10 \][/tex]
Now, divide each term by [tex]\(-2\)[/tex]:
[tex]\[ y = 3x - 5 \][/tex]
### Step 4: Identify the Solution Type
Since we were able to solve [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex], and knowing this represents a straight line with no restrictions causing inconsistency, the equation can have any number of [tex]\(x\)[/tex] values paired with corresponding [tex]\(y\)[/tex] values along the line [tex]\(y = 3x - 5\)[/tex].
### Conclusion
Given there are no constraints that make the equation inconsistent, and it describes a line in a two-dimensional plane, this line has infinitely many points.
Thus, the correct answer is:
(D) Infinitely Many solutions
### Step 1: Identify the Type of Equation
The given equation [tex]\(6x - 2y = 10\)[/tex] is a linear equation in two variables (x and y). A linear equation in two variables typically represents a straight line in a two-dimensional coordinate system.
### Step 2: Analyze the Structure of the Equation
A linear equation in the form [tex]\(Ax + By = C\)[/tex] can have different types of solutions based on its structure:
1. Infinitely Many Solutions: This occurs when the same line is represented in different forms (i.e., the same equation can be rewritten in multiple ways).
2. 1 Solution: Typically, this applies in the context of systems of equations where two lines intersect at exactly one point.
3. No Solution: This happens if two lines are parallel and distinct, meaning they never intersect.
### Step 3: Solve for One Variable
To understand the solutions better, we can express [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex]:
[tex]\[ 6x - 2y = 10 \][/tex]
Solve for [tex]\( y \)[/tex] by subtracting [tex]\( 6x \)[/tex] from both sides:
[tex]\[ -2y = -6x + 10 \][/tex]
Now, divide each term by [tex]\(-2\)[/tex]:
[tex]\[ y = 3x - 5 \][/tex]
### Step 4: Identify the Solution Type
Since we were able to solve [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex], and knowing this represents a straight line with no restrictions causing inconsistency, the equation can have any number of [tex]\(x\)[/tex] values paired with corresponding [tex]\(y\)[/tex] values along the line [tex]\(y = 3x - 5\)[/tex].
### Conclusion
Given there are no constraints that make the equation inconsistent, and it describes a line in a two-dimensional plane, this line has infinitely many points.
Thus, the correct answer is:
(D) Infinitely Many solutions
