The problem in the image asks to solve the system of equations using three methods: elimination, substitution, and graphing the straight line. The given system is:

$x + 2y = 8$ $2x - y = 6$

Let's go through each method to solve the system.

1. Elimination Method

We aim to eliminate one variable by adding or subtracting the equations.

Step 1: Multiply the second equation by 2 to make the coefficients of $y$ in both equations equal.

$2(2x - y = 6) \Rightarrow 4x - 2y = 12$

Now, the system becomes: $x + 2y = 8$ $4x - 2y = 12$

Step 2: Add the two equations together to eliminate $y$:

$(x + 2y) + (4x - 2y) = 8 + 12$ $5x = 20$

Step 3: Solve for $x$:

$x = \frac{20}{5} = 4$

Step 4: Substitute $x = 4$ back into the first equation to find $y$:

$4 + 2y = 8$ $2y = 8 - 4$ $y = \frac{4}{2} = 2$

Thus, the solution is $x = 4$ and $y = 2$.

2. Substitution Method

Step 1: Solve one equation for $x$ or $y$. From the first equation, solve for $x$:

$x = 8 - 2y$

Step 2: Substitute this expression for $x$ into the second equation:

$2(8 - 2y) - y = 6$ $16 - 4y - y = 6$ $16 - 5y = 6$ $-5y = 6 - 16$ $-5y = -10$ $y = \frac{-10}{-5} = 2$

Step 3: Substitute $y = 2$ back into the equation $x = 8 - 2y$:

$x = 8 - 2(2) = 8 - 4 = 4$

So, the solution is again $x = 4$ and $y = 2$.

3. Graphing Method

For graphing, we need to rewrite the equations in slope-intercept form $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.

First equation: $x + 2y = 8 \Rightarrow 2y = -x + 8 \Rightarrow y = -\frac{1}{2}x + 4$

Second equation: $2x - y = 6 \Rightarrow -y = -2x + 6 \Rightarrow y = 2x - 6$

Now, plot these two lines:

1. The first line has a slope of $-\frac{1}{2}$ and a y-intercept of 4.
2. The second line has a slope of 2 and a y-intercept of $-6$.

The point where the two lines intersect will give the solution to the system. By graphing, the lines intersect at the point $(4, 2)$, confirming that $x = 4$ and $y = 2$ is the solution.

Would you like more details or have any questions?

Here are 5 related questions for deeper understanding:

1. How do you determine when to use elimination versus substitution?
2. What are the advantages of using the graphing method for solving systems of equations?
3. Can a system of linear equations have no solutions or infinitely many solutions? How can this be identified?
4. How can you verify if the obtained solution is correct for both equations?
5. What happens if both equations have the same slope but different y-intercepts?

Tip: When solving systems of equations, always check the solution by substituting the values back into both original equations to ensure they satisfy both.