Compare and contrast the following two equations using the [Cramer's rule] formula:
x + 2y = 7
2x + 4y = 14
Explain the similarities and differences between the two equations in terms of their coefficients and variables. Use the Cramer's rule formula to find the solution to the system of equations.
user
Sure, let's work through this problem.

First, let's identify the coefficients and variables in each equation:

Equation 1: y = (7 - 1 * x) / 2
   - This equation has two coefficients (1 and 2) and a single variable, y. The constant term is subtracted from x.

Equation 2: 2x = y - 7
   - This equation has two coefficients (2 and 1) and a single variable, x. The constant term is added to y.

Now, let's apply Cramer's rule, which is a method of solving systems of linear equations with three or more variables:

Cramer's rule for system of linear equations:
   - If |A| ≠ 0, then the unique solution is:
       x = |A22| / |A|
       y = |A12| / |A|
       z = |A11| / |A|

   Given system of linear equations:

       x + 2y = 7
       2x + 4y = 14

   The coefficient matrix, A, is:

       A = [1,2; 2,4]

   The determinant of A, denoted |A|, is:

       |A| = (2*4) - (2*2) = 0

   Since |A| = 0, the system is inconsistent, and we cannot use Cramer's rule to find the solution.

   However, for illustration purposes, let's proceed with the Cramer's rule calculation. The formula for the determinant of a 2x2 matrix is:

       |A| = ad - bc

   Using this formula, we calculate:

       |A| = (7)(4) - (2)(14) = -28

   To find x, we calculate the determinant of the matrix obtained by striking out the first row and first column:

       A22 = [14,14]

   Thus, |A22| = 14 * 14 = 196.

   To find y, we calculate the determinant of the matrix obtained by striking out the first column and first row:

       A12 = [2,14]

   Thus, |A12| = 2 * 14 = 28.

   Therefore, the unique solution is:

       x = |A22| / |A| = 196 / (-28) = -7

       y = |A12| / |A| = 28 / (-28) = 1

So, despite the fact that the system is inconsistent, we can still calculate the solution using Cramer's rule, although it won't be a real solution. In this case, the determinant of the coefficient matrix is zero, so the system is inconsistent, and this Cramer's rule calculation will produce complex or ambiguous results. Nonetheless, it was a valid exercise in showing you how to use the formulas.
assistant
Compare and contrast the following two equations using the [Cramer's rule] formula:
x + 2y = 7
2x + 4y = 14
Explain the similarities and differences between the two equations in terms of their coefficients and variables. Use the Cramer's rule formula to find the solution to the system of equations.
user
Sure, let's first understand what Cramer's rule is. Cramer's rule is a method for solving a system of linear equations with as many equations as unknowns, using determinants. It's named after Gabriel Cramer who published the rule for an arbitrary number of unknowns in 1750.

Now, let's look at the two equations:

1) x + 2y = 7
2) 2x + 4y = 14

The coefficients of the second equation are exactly twice those of the first equation, and the constant term on the right side of the second equation is also twice that of the first equation. This means that the second equation is a scalar multiple of the first one. In other words, they are dependent equations and represent the same line in a two-dimensional space.

When we try to solve this system using Cramer's rule, we'll see that the determinant of the coefficient matrix is zero, which confirms that the system does not have a unique solution.

The coefficient matrix is:
a11 = 1, a12 = 2
a21 = 2, a22 = 4

The determinant of this matrix (D) is:
D = a11*a22 - a12*a21 = 1*4 - 2*2 = 0

Since the determinant is zero, we can't apply Cramer's rule to find a unique solution. This is consistent with the fact that the two equations represent the same line and thus if we treat them as a system, it has an infinite number of solutions.

In conclusion, the two equations are similar in that they represent the same line, and different in that the coefficients and constants of the second equation are twice those of the first. When trying to solve this system using Cramer's rule, we find that the determinant of the coefficient matrix is zero, which means the system does not have a unique solution.
assistant