Questions: Solve the system by substitution. If the system is inconsistent or has dependent equations, say so. 2x - 9y = -13 -2x + y = 0 Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The system has a single solution. The solution set is . (Type an ordered pair. Type integers or simplified fractions.) B. There are infinitely many solutions and the equations are dependent. The solution set is (x, y) -2x+y=0. C. The system is inconsistent. The solution set is the empty set.

$Solve the system by substitution. If the system is inconsistent or has dependent equations, say so. 2x - 9y = -13 -2x + y = 0 Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The system has a single solution. The solution set is . (Type an ordered pair. Type integers or simplified fractions.) B. There are infinitely many solutions and the equations are dependent. The solution set is (x, y) -2x+y=0. C. The system is inconsistent. The solution set is the empty set.$

Transcript text: Solve the system by substitution. If the system is inconsistent or has dependent equations, say so. \[ \begin{aligned} 2 x-9 y & =-13 \\ -2 x+y & =0 \end{aligned} \] Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The system has a single solution. The solution set is $\square$ \}. (Type an ordered pair. Type integers or simplified fractions.) B. There are infinitely many solutions and the equations are dependent. The solution set is $\{(x, y) \mid-2 x+y=0\}$. C. The system is inconsistent. The solution set is the empty set.

Solution

Solution Steps

To solve the system by substitution, first solve one of the equations for one variable. Substitute this expression into the other equation to find the value of one variable. Then, use this value to find the other variable. Check if the solution satisfies both equations.

Step 1: Solve for $ y $ in the Second Equation

Given the equation: \[ -2x + y = 0 \] Solve for $ y $: \[ y = 2x \]

Step 2: Substitute $ y = 2x $ into the First Equation

Substitute $ y = 2x $ into the first equation: \[ 2x - 9(2x) = -13 \] Simplify: \[ 2x - 18x = -13 \] \[ -16x = -13 \] Solve for $ x $: \[ x = \frac{13}{16} \]

Step 3: Solve for $ y $ Using $ x = \frac{13}{16} $

Substitute $ x = \frac{13}{16} $ into $ y = 2x $: \[ y = 2 \left(\frac{13}{16}\right) = \frac{26}{16} = \frac{13}{8} \]