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How many solutions does the system have?

{[3x+y=8],[2x+2y=8]:}
Choose 1 answer:
(A) Exactly one solution
(B) No solutions
(c) Infinitely many solutions

How many solutions does the system have? $\newline$ $\left\{\begin{array}{l} 3 x+y=8 \\ 2 x+2 y=8 \end{array}\right.$ $\newline$ Choose $1$ answer: $\newline$ (A) Exactly one solution $\newline$ (B) No solutions $\newline$ (C) Infinitely many solutions

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Find the number of solutions to a system of equations

Full solution

Q. How many solutions does the system have?

\newline

\left\{\begin{array}{l} 3 x+y=8 \\ 2 x+2 y=8 \end{array}\right.

\newline

Choose

1

answer:

\newline

(A) Exactly one solution

\newline

(B) No solutions

\newline

(C) Infinitely many solutions

Write down the system of equations: Let's first write down the system of equations: $\newline$ $3x + y = 8$ ...( $1$ ) $\newline$ $2x + 2y = 8$ ...( $2$ ) $\newline$ We will try to solve this system using the method of substitution or elimination.
Simplify equation ( $2$ ): Let's simplify equation ( $2$ ) by dividing every term by $2$ to see if it becomes similar to equation ( $1$ ): $\newline$ $x + y = 4 \ ...(3)$ $\newline$ Now we have a new system of equations: $\newline$ $3x + y = 8 \ ...(1)$ $\newline$ $x + y = 4 \ ...(3)$
Subtract equation ( $3$ ) from equation ( $1$ ): We can subtract equation ( $3$ ) from equation ( $1$ ) to eliminate y and solve for x: $\newline$ $(3x + y) - (x + y) = 8 - 4$ $\newline$ $3x - x + y - y = 4$ $\newline$ $2x = 4$ $\newline$ $x = 2$
Solve for x: Now that we have the value of $x$ , we can substitute it back into either equation ( $1$ ) or ( $3$ ) to find the value of $y$ . Let's use equation ( $3$ ): $x + y = 4$ $2 + y = 4$ $y = 4 - 2$ $y = 2$
Substitute $x$ back into equation ( $3$ ): We have found a solution $(x, y) = (2, 2)$ that satisfies both equations. This means the system of equations has exactly one solution.

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