The system of equations given is:

1. $2x - y = 5$
2. $x + y = 4$

We will solve this system using the following methods:

1. Substitution Method
2. Elimination Method
3. Graphical Method

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1. Substitution Method

From the second equation $x + y = 4$, solve for $y$: $y = 4 - x$

Substitute $y = 4 - x$ into the first equation $2x - y = 5$: $2x - (4 - x) = 5$ $2x - 4 + x = 5$ $3x = 9$ $x = 3$

Substitute $x = 3$ back into $y = 4 - x$: $y = 4 - 3 = 1$

Solution: $x = 3, y = 1$

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2. Elimination Method

The system is:

1. $2x - y = 5$
2. $x + y = 4$

Add the two equations to eliminate $y$: $(2x - y) + (x + y) = 5 + 4$ $3x = 9$ $x = 3$

Substitute $x = 3$ into the second equation $x + y = 4$: $3 + y = 4$ $y = 1$

Solution: $x = 3, y = 1$

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3. Graphical Method

To graph these equations:

• Rewrite the first equation $2x - y = 5$ as $y = 2x - 5$.
• Rewrite the second equation $x + y = 4$ as $y = -x + 4$.

The solution is the intersection of the lines:

• $y = 2x - 5$
• $y = -x + 4$

Plot:

• Line 1: $y = 2x - 5$ passes through $(0, -5)$ and $(3, 1)$.
• Line 2: $y = -x + 4$ passes through $(0, 4)$ and $(3, 1)$.

The intersection point is $(3, 1)$, confirming $x = 3, y = 1$.

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Final Solution:

The solution to the system is: $x = 3, \, y = 1$

Would you like a graphical plot or further explanation for any method?

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Related Questions:

1. What are the advantages of the substitution method for solving systems of equations?
2. How can you verify the solution to a system of equations?
3. Can all systems of equations be solved graphically? Why or why not?
4. How would the solution change if the equations were $2x - y = 5$ and $x + y = 5$?
5. What is the general form of a linear equation and how does it help in graphing?

Tip:

Always check your solution by substituting the values of $x$ and $y$ back into the original equations!