Finding m for Infinite Solutions of Linear Equations

The given system of linear equations is:

1. $17x + my + 102 = 0$
2. $23x + 299y + 138 = 0$

We need to find the value of 'm' for which this system has an infinite number of solutions. A system of two linear equations in two variables, say $a_1x + b_1y + c_1 = 0$ and $a_2x + b_2y + c_2 = 0$, has infinitely many solutions if the ratio of their corresponding coefficients is equal. That is, the following condition must be satisfied:

$\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$

Let's identify the coefficients from our given equations:

• From Equation 1: $a_1 = 17$, $b_1 = m$, $c_1 = 102$
• From Equation 2: $a_2 = 23$, $b_2 = 299$, $c_2 = 138$

Now, let's set up the ratios of these coefficients:

$\frac{17}{23} = \frac{m}{299} = \frac{102}{138}$

For the system to have infinite solutions, all three ratios must be equal. Let's first simplify the ratio of the constant terms, $\frac{102}{138}$.

We can divide both the numerator and the denominator by their greatest common divisor. Both are even numbers, so we can start by dividing by 2:

$\frac{102 \div 2}{138 \div 2} = \frac{51}{69}$

Now, let's look at 51 and 69. Both are divisible by 3:

$\frac{51 \div 3}{69 \div 3} = \frac{17}{23}$

So, the simplified ratio of constant terms is $\frac{17}{23}$. This matches the ratio of the coefficients of 'x'.

Now, we use the condition $\frac{b_1}{b_2} = \frac{c_1}{c_2}$ (or $\frac{b_1}{b_2} = \frac{a_1}{a_2}$, since all ratios are equal) to find the value of 'm'.

$\frac{m}{299} = \frac{17}{23}$

To solve for 'm', multiply both sides of the equation by 299:

$m = \frac{17}{23} \times 299$

We can simplify this calculation by dividing 299 by 23 before multiplying by 17. Let's perform the division:

$\frac{299}{23} = 13$

So, the equation becomes:

$m = 17 \times 13$

Now, we calculate the product of 17 and 13:

$17 \times 13 = 17 \times (10 + 3) = 17 \times 10 + 17 \times 3 = 170 + 51 = 221$

Thus, the value of 'm' is 221.

Let's verify the ratios with $m = 221$:

Since $\frac{17}{23} = \frac{221}{299} = \frac{102}{138} = \frac{17}{23}$, the condition for infinite solutions is satisfied when $m = 221$.

Therefore, the value of m for which the given system of equations has an infinite number of solutions is 221.

Revision Table: Linear Equations and Solutions

Additional Information on Systems of Linear Equations

A system of linear equations is a set of two or more linear equations involving the same variables. In this case, we have a system of two linear equations in two variables, 'x' and 'y'.

Solving a system of linear equations means finding the values of the variables that satisfy all equations in the system simultaneously. The solution can be found using various methods:

• Substitution Method: Solve one equation for one variable and substitute that expression into the other equation.
• Elimination Method: Multiply one or both equations by constants so that the coefficients of one variable are opposites, then add the equations to eliminate that variable.
• Graphical Method: Graph both equations on the same coordinate plane. The intersection point (if any) represents the solution. For infinite solutions, the lines are the same; for no solution, the lines are parallel.
• Matrix Methods: Using augmented matrices and row operations (like Gaussian elimination) or determinants.

The type of solution (unique, infinite, or no solution) depends on the relationship between the lines represented by the equations, which is determined by the ratios of their coefficients.