Question

Graph the system below and write its solution. {; y=2x+1 x+2y=7 Note that you can also answer "No solution" or "Infinitely many" solutions.

Answer 1

To solve the system of equations graphically and determine its solution, we'll start with the two given equations:

1. \( y = 2x + 1 \) (Equation 1)
2. \( x + 2y = 7 \) (Equation 2)

**Step 1: Graphing Equation 1**

The first equation is in slope-intercept form (y = mx + b):
- Slope (m) = 2
- y-intercept (b) = 1

To graph this line:
- Start at the y-intercept (0, 1).
- From (0, 1), use the slope to find another point. Since the slope is 2, rise 2 and run 1 to get to the point (1, 3).
- Plot these points and draw a line through them.

**Step 2: Graphing Equation 2**

The second equation can be rearranged to the slope-intercept form. Start with the original equation:
- \( x + 2y = 7 \)
- Subtract \( x \) from both sides:
\[ 2y = -x + 7 \]
- Divide everything by 2:
\[ y = -\frac{1}{2}x + \frac{7}{2} \]

Now the equation is in slope-intercept form:
- Slope (m) = -1/2
- y-intercept (b) = 7/2

To graph this line:
- Start at the y-intercept (0, 7/2), which is approximately (0, 3.5).
- From (0, 7/2), use the slope to find another point. Since the slope is -1/2, drop down 1 and move right 2 to get to the point (2, 3).
- Plot these points and draw a line through them.

**Step 3: Finding the Intersection**

To find the solution to the system, we need to look for the point where these two lines intersect.

1. Equation 1: \( y = 2x + 1 \)
2. Substitute Equation 1 into Equation 2:
- \( x + 2(2x + 1) = 7 \)
- \( x + 4x + 2 = 7 \)
- \( 5x + 2 = 7 \)
- \( 5x = 5 \)
- \( x = 1 \)

3. Substitute \( x = 1 \) back into Equation 1 to find \( y \):
- \( y = 2(1) + 1 = 3 \)

Thus, the point of intersection and the solution to the system is \( (1, 3) \).

**Final Answer:**
- The solution to the system is \( (1, 3) \).