Questions: Solve the given system by the substitution method. If there is no solution or an infinite number of solutions, use set notation to express solution sets. x+3y=6 2x+3y=9 Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. There is exactly one solution. The solution set is 3. (Type an ordered pair, using integers or fractions. Simplify your answer.) B. There are infinitely many solutions. The solution set is (x, y) x+3y=6 or (x, y) 2x+3y=9. C. There is no solution. The solution set is the empty set, ∅.

Solve the given system by the substitution method. If there is no solution or an infinite number of solutions, use set notation to express solution sets. x+3y=6 2x+3y=9 Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. There is exactly one solution. The solution set is 3. (Type an ordered pair, using integers or fractions. Simplify your answer.) B. There are infinitely many solutions. The solution set is (x, y) x+3y=6 or (x, y) 2x+3y=9. C. There is no solution. The solution set is the empty set, ∅.
Transcript text: Solve the given system by the substitution method. If there is no solution or an infinite number of solutions, so s Use set notation to express solution sets. \[ \left\{\begin{array}{l} x+3 y=6 \\ 2 x+3 y=9 \end{array}\right. \] Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. There is exactly one solution. The solution set is $\square$ 3. (Type an ordered pair, using integers or fractions. Simplify your answer.) B. There are infinitely many solutions. The solution set is $\{(x, y) \mid x+3 y=6\}$ or $\{(x, y) \mid 2 x+3 y=9\}$. C. There is no solution. The solution set is the empty set, $\varnothing$.
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Solution

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Solution Steps

Step 1: Calculate the determinant of the coefficient matrix

The determinant is calculated as \(det = a1 \times b2 - a2 \times b1 = 1 \times 3 - 2 \times 3 = -3\).

Step 2: Apply Cramer's Rule to find the values of x and y

For x, the determinant is calculated as \(det_x = c1 \times b2 - c2 \times b1 = 6 \times 3 - 9 \times 3 = -9\). For y, the determinant is calculated as \(det_y = a1 \times c2 - a2 \times c1 = 1 \times 9 - 2 \times 6 = -3\).

Step 3: Calculate the values of x and y

x is calculated as \(x = \frac{det_x}{det} = \frac{-9}{-3} = 3\). y is calculated as \(y = \frac{det_y}{det} = \frac{-3}{-3} = 1\).

Final Answer: The solution to the system of equations is x = 3 and y = 1.

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