Solve the system by graphing.

left brace Start 2 By 1 Matrix 1st Row 1st Column 2 x plus y equals 2 2nd Row 1st Column y equals 4 x minus 7 EndMatrix

To solve the system of equations by graphing, we need to graph both equations on the same coordinate plane and find the point of intersection.

The system of equations is:
\[ \begin{cases} 2x + y = 2 \\ y = 4x - 7 \end{cases} \]

### Step 1: Graph the First Equation

Rewrite the first equation \(2x + y = 2\) in slope-intercept form \(y = mx + b\).

\[ 2x + y = 2 \]
\[ y = -2x + 2 \]

This equation has a slope of \(-2\) and a y-intercept of \(2\).

### Step 2: Graph the Second Equation

The second equation is already in slope-intercept form:

\[ y = 4x - 7 \]

This equation has a slope of \(4\) and a y-intercept of \(-7\).

### Step 3: Plot Both Equations on the Coordinate Plane

1. For \(y = -2x + 2\):
   - The y-intercept is \(2\), so plot the point \((0, 2)\).
   - The slope is \(-2\), which means for every 1 unit increase in \(x\), \(y\) decreases by 2 units. From \((0, 2)\), move 1 unit to the right (to \(x = 1\)) and 2 units down (to \(y = 0\)). Plot the point \((1, 0)\).

2. For \(y = 4x - 7\):
   - The y-intercept is \(-7\), so plot the point \((0, -7)\).
   - The slope is \(4\), which means for every 1 unit increase in \(x\), \(y\) increases by 4 units. From \((0, -7)\), move 1 unit to the right (to \(x = 1\)) and 4 units up (to \(y = -3\)). Plot the point \((1, -3)\).

### Step 4: Find the Intersection Point

By plotting both lines on the same graph, we can find their intersection point. Let's calculate it algebraically to confirm.

Set the equations equal to each other since both equal \(y\):

\[ -2x + 2 = 4x - 7 \]

Solve for \(x\):

\[ -2x + 2 = 4x - 7 \]
\[ 2 + 7 = 4x + 2x \]
\[ 9 = 6x \]
\[ x = \frac{9}{6} = \frac{3}{2} \]

Now, substitute \(x = \frac{3}{2}\) back into either equation to find \(y\). Using \(y = 4x - 7\):

\[ y = 4\left(\frac{3}{2}\right) - 7 \]
\[ y = 6 - 7 \]
\[ y = -1 \]

The intersection point is \(\left(\frac{3}{2}, -1\right)\).

### Conclusion

The solution to the system of equations is the point \(\left(\frac{3}{2}, -1\right)\). This is where the two lines intersect when graphed.

Would you like more details or have any questions?

Here are 5 questions you might want to ask next:
1. How do you solve a system of equations using the substitution method?
2. What is the elimination method for solving systems of equations?
3. Can you solve a system of nonlinear equations by graphing?
4. How can you determine if a system of equations has no solution or infinitely many solutions?
5. Can you explain how to use matrices to solve systems of equations?

**Tip:** When graphing lines, make sure to carefully plot the intercepts and use the slope to determine additional points for accuracy.

Learn how to solve a system of equations by graphing with detailed steps. This problem involves graphing two linear equations, finding their intersection point, and confirming algebraically. Suitable for high school students in Grades 9-12.

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