The system of equations 2x + y = 5; x - 3y = -1 and 3x + 4y = k is consistent, then k is equal to :

Question

Accepted Answer

To determine why Option 4 (k = 10) is the correct answer for the given system of equations to be consistent, we need to analyze the system of equations:

1. $2x + y = 5$
2. $x - 3y = -1$
3. $3x + 4y = k$

A system of equations is consistent if there is at least one set of values for the variables that satisfies all the equations simultaneously. Let's solve the first two equations to find the values of $x$ and $y$.

**Step 1: Solve the first two equations**

From equation (2): $x - 3y = -1$

We can express $x$ in terms of $y$:
$$ x = 3y - 1 $$

Substitute $x = 3y - 1$ into equation (1):
$$ 2(3y - 1) + y = 5 $$
$$ 6y - 2 + y = 5 $$
$$ 7y - 2 = 5 $$
$$ 7y = 7 $$
$$ y = 1 $$

Now substitute $y = 1$ back into the expression for $x$:
$$ x = 3(1) - 1 $$
$$ x = 3 - 1 $$
$$ x = 2 $$

So, the solution to the first two equations is $x = 2$ and $y = 1$.

**Step 2: Determine the value of $k$ for consistency**

Substitute $x = 2$ and $y = 1$ into equation (3):
$$ 3x + 4y = k $$
$$ 3(2) + 4(1) = k $$
$$ 6 + 4 = k $$
$$ k = 10 $$

Thus, for the system to be consistent, $k$ must be equal to 10, which corresponds to Option 4.

**Explanation of Incorrect Options:**

- **Option 1 (k = 1):** If $k = 1$, substituting $x = 2$ and $y = 1$ into equation (3) would not satisfy the equation, as $3(2) + 4(1) = 10 \neq 1$. Therefore, this option does not make the system consistent.

- **Option 2 (k = 2):** Similarly, if $k = 2$, substituting $x = 2$ and $y = 1$ into equation (3) would not satisfy the equation, as $3(2) + 4(1) = 10 \neq 2$. Thus, this option is incorrect.

- **Option 3 (k = 5):** If $k = 5$, substituting $x = 2$ and $y = 1$ into equation (3) would not satisfy the equation, as $3(2) + 4(1) = 10 \neq 5$. Therefore, this option is also incorrect.

In conclusion, Option 4 (k = 10) is the only value that makes the system of equations consistent, as it satisfies all three equations simultaneously.