Exercise 31 Page 202

We want to solve the given system of equations using matrices to find the point of intersection of the lines. We need to write the system as a matrix equation. y-x=5 2y-2x=8 ⇒ [ c y-x 2y-2x ] = [ c 5 8 ] Now, we will write the left-hand side of the above matrix equation as the product of the coefficient matrix and the variable matrix. [ c y-x 2y-2x ] = [ c 5 8 ] ⇕ [ cc 1 & -1 2 & -2 ] * [ c y x ] = [ c 5 8 ]We will solve the matrix equation by multiplying both sides by the inverse of the coefficient matrix to isolate the x and y variables. When multiplying a matrix by its inverse, we obtain the identity matrix. To find the inverse of a 2* 2 matrix, we use the corresponding formula.

The expression ad-bc is known as the determinant of a 2* 2 matrix. Because it is in the denominator of a fraction, if the determinant is zero, the matrix cannot have an inverse. Consider our coefficient matrix. [ cc 1 & -1 2 & -2 ] Let's calculate its determinant.

Since the determinant is zero, the matrix does not have an inverse. Therefore, the system has either no solution or infinitely many solutions. To choose one of these options we need to use an alternative method. Let's go back to our equations and write both of them in the slope-intercept form.

Comparing the slopes, we see that they are equal, so the lines are either parallel or the same. Looking at the y-intercepts, we can tell the lines are different because the point at which each line crosses the y-axis is different. This means that the lines are parallel and the system has no solution.