Types of Solutions for a System

Solve the given system of equations. Choose the best answer.

{
2
𝑥
+
𝑦
=
4

8
𝑥
+
4
𝑦
=
20

Question

Types of Solutions for a System

Solve the given system of equations. Choose the best answer.

{
2
𝑥
+
𝑦
=
4

8
𝑥
+
4
𝑦
=
20

BlackTom AI · Accepted Answer

Let's examine what each option says by analyzing the given system step by step.
Option 1: There is one solution only to the system. To test this, we can compare the two equations. If we multiply the first equation by 4, we get 8x + 4y = 16. This is inconsistent with the second equation, which is 8x + 4y = 20. Since 16 ≠ 20, the two lines are parallel and do not intersect, indicating there is not exactly one solution. Therefore, this option is incorrect.
Option 2: There are infinitely many solutions to the system. Infinitely many solutions would occur if the two equations represented the same line, i.e., were multiples of each other. However, as shown, multiplying the first equation by 4 yields 8x + 4y = 16, not 20 on the right-hand side. The mismatch shows the lines are distinct parallel lines, not the same line, so this option is false.
Option 3: There is no solution to the system. This would be the case when the two equations represent parallel lines that never meet. Since 8x + 4y cannot equal both 16 and 20 simultaneously, the system has no point in common. This is the correct characterization for this pair of equations.
In summary, the first equation and the scaled version of it imply 8x + 4y must equal 16, but the second equation requires 8x + 4y to equal 20. The contradiction confirms there is no solution, ruling out both having a unique solution and having infinitely many solutions.

MAT-265-OD24-02-7799 Prior Knowledge Check

Types of Solutions for a System Solve the given system of equations. Choose the best answer. { 2 𝑥 + 𝑦 = 4 8 𝑥 + 4 𝑦 = 20

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