Finding the Value of m for No Solution in Linear Equations

We are given a system of two linear equations in two variables, x and y. We need to find the value of the parameter 'm' such that this system has no solution.

The given system of equations is:

1. \(18x - 72y + 13 = 0\)
2. \(7x - my - 17 = 0\)

A general system of two linear equations in two variables is represented as:

\(a_1x + b_1y + c_1 = 0\)

\(a_2x + b_2y + c_2 = 0\)

By comparing the given equations with the general form, we can identify the coefficients:

From equation (1):

• \(a_1 = 18\)
• \(b_1 = -72\)
• \(c_1 = 13\)

From equation (2):

• \(a_2 = 7\)
• \(b_2 = -m\)
• \(c_2 = -17\)

For a system of linear equations to have no solution, the lines represented by the equations must be parallel and distinct. This condition is mathematically expressed as the ratio of the coefficients of x and y being equal, but not equal to the ratio of the constant terms:

\(\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}\)

Let's apply this condition to the given system of equations.

First, consider the equality of the ratios of x and y coefficients:

\(\frac{a_1}{a_2} = \frac{b_1}{b_2}\)

Substitute the identified coefficients:

\(\frac{18}{7} = \frac{-72}{-m}\)

\(\frac{18}{7} = \frac{72}{m}\)

Now, we can solve this equation for 'm'. We can cross-multiply:

\(18 \times m = 7 \times 72\)

\(18m = 504\)

Divide both sides by 18:

\(m = \frac{504}{18}\)

\(m = 28\)

Next, we need to verify that for \(m = 28\), the ratio of the coefficients of y is not equal to the ratio of the constant terms. That is, we must check if:

\(\frac{b_1}{b_2} \neq \frac{c_1}{c_2}\)

Substitute the values with \(m = 28\):

\(\frac{-72}{-m} = \frac{-72}{-28} = \frac{72}{28} = \frac{18 \times 4}{7 \times 4} = \frac{18}{7}\)

And the ratio of constant terms is:

\(\frac{c_1}{c_2} = \frac{13}{-17}\)

Clearly, \(\frac{18}{7} \neq \frac{13}{-17}\). The condition for no solution is satisfied when \(m = 28\).

Therefore, the system of equations will have no solution when the value of m is 28.

Revision Table: System of Linear Equations Conditions

Additional Information on Linear Equations and Solutions

A system of linear equations represents lines in a coordinate plane. The solution(s) to the system are the point(s) where these lines intersect.

• Unique Solution: When the lines intersect at exactly one point. This happens when their slopes are different.
• No Solution: When the lines are parallel and never intersect. This happens when they have the same slope but different y-intercepts. The condition \(\frac{a_1}{a_2} = \frac{b_1}{b_2}\) implies the slopes are equal (since slope \( = -a/b\)), and \(\frac{b_1}{b_2} \neq \frac{c_1}{c_2}\) implies the y-intercepts are different.
• Infinitely Many Solutions: When the lines are coincident, meaning they are the same line. This happens when they have the same slope and the same y-intercept.

Understanding these geometric interpretations helps in grasping the algebraic conditions for the number of solutions a system of linear equations can have.