Solve the system using graphical method:    
	{■( 4x+6y=2@2x-y=5)┤


To solve the system of equations graphically, we will follow these steps:

Given system of equations:
\[
\begin{aligned}
4x + 6y &= 2 \quad \text{(1)} \\
2x - y &= 5 \quad \text{(2)}
\end{aligned}
\]

### Step 1: Rewrite both equations in slope-intercept form \(y = mx + b\).

#### For Equation (1): \(4x + 6y = 2\)
Solve for \(y\):
\[
6y = -4x + 2
\]
\[
y = \frac{-4}{6}x + \frac{2}{6}
\]
\[
y = -\frac{2}{3}x + \frac{1}{3}
\]
So, the equation is:
\[
y = -\frac{2}{3}x + \frac{1}{3}
\]

#### For Equation (2): \(2x - y = 5\)
Solve for \(y\):
\[
-y = -2x + 5
\]
\[
y = 2x - 5
\]
So, the equation is:
\[
y = 2x - 5
\]

### Step 2: Graph both equations.

- The first equation \(y = -\frac{2}{3}x + \frac{1}{3}\) has a slope of \(-\frac{2}{3}\) and a y-intercept at \(\frac{1}{3}\).
- The second equation \(y = 2x - 5\) has a slope of \(2\) and a y-intercept at \(-5\).

### Step 3: Plot the lines on a graph and find their point of intersection.

- Line 1 (\(y = -\frac{2}{3}x + \frac{1}{3}\)) will be a decreasing line, crossing the y-axis at \(\frac{1}{3}\).
- Line 2 (\(y = 2x - 5\)) will be an increasing line, crossing the y-axis at \(-5\).

The solution to the system will be the point where these two lines intersect.

### Step 4: Solve algebraically for verification.

We can solve the system of equations algebraically to verify the intersection point.

From the second equation:
\[
y = 2x - 5
\]
Substitute this into the first equation:
\[
4x + 6(2x - 5) = 2
\]
\[
4x + 12x - 30 = 2
\]
\[
16x = 32
\]
\[
x = 2
\]
Substitute \(x = 2\) into the second equation to find \(y\):
\[
y = 2(2) - 5 = 4 - 5 = -1
\]

### Final Answer:
The solution to the system is \(x = 2\), \(y = -1\). So the lines intersect at the point \((2, -1)\).

Would you like a graphical representation of this solution or any further details?

### Related questions:
1. What is the graphical interpretation of the slope and y-intercept?
2. How can you check if two lines are parallel or perpendicular?
3. What is the significance of the intersection point in solving systems graphically?
4. How would this solution change if the equations were nonlinear?
5. Can a system of linear equations have no solution or infinite solutions graphically?

**Tip:** Always check the algebraic solution to confirm the point of intersection before concluding graphical solutions!

Solve the system using graphical method: {4x + 6y = 2, 2x - y = 5}

Explore how to solve a system of linear equations graphically with the example 4x + 6y = 2 and 2x - y = 5. Learn the step-by-step process, including rewriting the equations in slope-intercept form, graphing the lines, and finding the intersection point. Suitable for high school students in Grades 9-12.

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