Questions: Question 4 1 pts Solve the system. x+3 y=9 2 x+6 y=18 (0,0) y=-x/3+3, where x is any real number (9, 0) inconsistent (no solution)

Transcript text: Question 4 1 pts Solve the system. \[ \left\{\begin{array}{c} x+3 y=9 \\ 2 x+6 y=18 \end{array}\right. \] $(0,0)$ $y=-\frac{x}{3}+3$, where $x$ is any real number (9, 0) inconsistent (no solution)

Solution

Solution Steps

Step 1: Analyze the System of Equations

The given system of equations is:

\[ \begin{align*}

& \quad x + 3y = 9 \\
& \quad 2x + 6y = 18 \end{align*} \]

Step 2: Simplify the Second Equation

Notice that the second equation can be simplified by dividing every term by 2:

\[ 2x + 6y = 18 \quad \Rightarrow \quad x + 3y = 9 \]

Step 3: Compare the Equations

After simplification, both equations are identical:

\[ x + 3y = 9 \]

This means that the two equations represent the same line. Therefore, the system has infinitely many solutions, which can be expressed in terms of one of the variables.

Step 4: Express the Solution

We can express $ y $ in terms of $ x $ from the equation:

\[ x + 3y = 9 \quad \Rightarrow \quad 3y = 9 - x \quad \Rightarrow \quad y = -\frac{x}{3} + 3 \]

This means that for any real number $ x $, $ y $ can be calculated using the equation $ y = -\frac{x}{3} + 3 $.