91Ó°ÊÓ

The substitution method is a straightforward way to solve a system of linear equations. It involves solving one of the equations for one variable and then substituting that expression into the other equation. This method is especially useful when one of the equations is already solved for a single variable.

In our exercise, notice that both given equations are already solved for y:
1) \[y = \frac{2}{5}x - 7 \]2) \[y = \frac{2}{5}x + 4 \]

To use the substitution method, we set these two equations equal to each other because they both represent y:
\[ \frac{2}{5}x - 7 = \frac{2}{5}x + 4 \]
Next, we need to isolate one of the variables. By subtracting \( \frac{2}{5}x \) from both sides, we end up with:
\[ -7 = 4 \]
Since this statement is false, it indicates there is no solution. This outcome tells us the lines represented by the equations are parallel and do not intersect.

The elimination method is another powerful technique for solving systems of linear equations. This method involves adding or subtracting the equations to cancel out one of the variables, making it easier to solve for the other variable.

Given the equations in our exercise:
1) \[y = \frac{2}{5}x - 7 \]2) \[y = \frac{2}{5}x + 4 \]
We can rewrite them in a way that might make elimination simpler. Essentially, we aim to align the equations such that either x or y can be eliminated. However, these equations are already simplified in terms of y.
Setting them equal, as seen in the substitution method, results in:
\[ \frac{2}{5}x - 7 = \frac{2}{5}x + 4 \]

Subtracting \( \frac{2}{5}x \) from both sides gives:
\[ -7 = 4 \]
This shows that elimination does not yield a solvable equation, confirming that the system has no solution because the lines are parallel and never intersect.

The graphing method involves plotting each equation on a coordinate plane and identifying the point(s) where the lines intersect. Let's discuss how we can use this method to understand our given system of equations.

The given equations are:
1) \[y = \frac{2}{5}x - 7 \]2) \[y = \frac{2}{5}x + 4 \]

To plot these lines, we first identify their slopes and y-intercepts.
The first equation has a slope of \[ \frac{2}{5} \] and a y-intercept of -7. The second equation also has a slope of \[ \frac{2}{5} \] but with a y-intercept of 4. These slopes being equal means the lines are parallel.

When two lines are parallel, they never meet. This visual representation aligns with our analytical conclusion from substitution and elimination methods: the system of equations has no solution. The graph confirms that the lines do not intersect, providing a visual proof of the inconsistency of the system.