Solution Of A Linear Equation

To solve the given equation , we need to simplify and solve for .

1. Start with the given equation:

2. Remove the parentheses on the left-hand side:

3. Combine like terms:
Left-hand side:
Right-hand side:

4. Now set the simplified expressions equal:

5. Move all terms to one side:
Add to both sides:

6. Solve for :
Add 4 to both sides:

Final Answer:
Option (C) 2.25 is correct.

To solve this problem, we need to find the solution to the system of linear equations:

1. Solve the system using elimination or substitution method.

Step 1: Multiply both equations to eliminate one variable:
Multiply equation (1) by 3 and equation (2) by 4 to make coefficients of equal in magnitude:

Step 2: Add equations (3) and (4):

Step 3: Substitute into one of the original equations:
Using equation (1):

So the solution is:

2. Evaluating the options:
Only one option is not equal to (1, -1):

• (1) → Not equal ✔
• (2) → Equal
• (3) → Equal
• (4) → Equal

Final Answer:
The correct answer is (A) as it is not equal to the solution .

To solve this system of equations and find the value of , we start with the given:

1. Given equations:
(1)
(2)

2. Solve Equation (1) for :
From (1):

3. Substitute into Equation (2):
Substitute into equation (2):

4. Simplify the equation:

5. Solve for :
Subtract 49 from both sides:

Final Answer:
Option (A) 1 is correct.

To solve this question, we need to identify which of the given equations is not a linear equation.

1. Understanding Linear Equations:
A linear equation in two variables is an equation of the form:
where , , and are constants, and the powers of the variables and are both 1.

2. Checking Each Option:

• (1) → Rearranged: → Linear ✔
• (2) → Rearranged: → Contains , so not linear ❌
• (3) → Rearranged: → Linear ✔
• (4) → Already in linear form ✔

Final Answer:
The correct answer is (B) because it contains , which makes it a nonlinear equation.

To solve the problem, we need to determine the condition under which the given system of linear equations has a unique solution. The system of equations is:

$ , where is the slope.

For the first equation :

\) .

For the second equation :

\) .

3. Ensuring the Lines Are Not Parallel:
For the system to have a unique solution, the slopes and must be different:

\) and :

\) :
Multiply both sides by 2 to eliminate the denominators:

\) $

5. Conclusion:
The condition for the system to have a unique solution is that .

Final Answer:
The correct option is .

To solve the problem, we need to find the value of given the system of linear equations:

Step 1: Solve for in terms of from the second equation. From the second equation: Subtract from both sides: Step 2: Substitute into the first equation. The first equation is: Substitute : Step 3: Simplify the equation. Distribute the : Combine like terms: Step 4: Solve for . Add 20 to both sides: Divide by 22: Step 5: Verify the solution. Substitute back into the expression for : Now check both equations: 1. For : (True) 2. For : (True) Both equations are satisfied, so the solution is correct.

Final Answer:

To solve the problem, we need to analyze the pair of linear equations and determine their geometric relationship or the number of solutions they have.

1. Given Equations:
Equation (1):
Equation (2):

2. Convert Both Equations to Standard Form:
Multiply equation (1) by 6 to eliminate denominators:

⇒

Now, the system becomes:
Equation (1):
Equation (2):

3. Solve the System of Equations:
Add the two equations:

⇒
⇒

Now substitute in Equation (1):

⇒
⇒
⇒

4. Conclusion:
The pair of equations has a unique solution: .
Therefore, the lines intersect at a single point.

Final Answer:
The system of equations has one solution.

To solve the problem, we analyze the relationship between two linear equations based on their coefficients.

1. General Form of a Pair of Linear Equations:
Consider two linear equations in the form:

2. Condition Given:
It is given that:

3. Interpretation of the Condition:
This means that the two lines have the same slope (they are parallel), but different intercepts.
Therefore, the lines are parallel and **distinct**.

4. Geometrical Implication:
Since the lines are parallel but not the same, they will never intersect.
Thus, the system of equations has no solution.

Final Answer:
The lines are such that there are

To solve the problem, we need to analyze the system of two equations: and , and determine how many solutions this system has.

1. Understanding the Given Equations:
The two equations given are:

2. Analyzing the Equations:
These are two linear equations representing vertical lines in the coordinate plane. The line is the y-axis, and the line is a vertical line 5 units to the right of the y-axis.

3. Intersections of the Lines:
Since these lines are parallel (both vertical) and have different x-values, they will never intersect. Hence, there is no common solution.

Final Answer:
The system of equations has no solution.