To solve the system of equations by graphing, we'll first graph each equation.
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Equation 1: \(y = 2x + 3\)
- This is a linear equation with a slope of 2 and a y-intercept of 3.
- Plot the y-intercept (0, 3) on the graph.
- From there, use the slope (rise/run) of 2 to find another point. For every 1 unit you move to the right (positive direction on the x-axis), move up 2 units (positive direction on the y-axis).
- The next point is (1, 5).
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Equation 2: \(y = 2x + 1\)
- This is also a linear equation with a slope of 2 but a y-intercept of 1.
- Plot the y-intercept (0, 1) on the graph.
- Use the slope of 2 to find another point: from (0, 1), move right 1 unit and up 2 units to get the point (1, 3).
Now, we can plot both lines on the graph:
- The first line passes through points (0, 3) and (1, 5).
- The second line passes through points (0, 1) and (1, 3).
Observing the Graph:
When you graph these two equations, you'll notice that the two lines have the same slope (2) but different y-intercepts. This indicates that the lines are parallel and will never intersect.
Conclusion:
Since the lines are parallel, there is no solution to this system of equations; the system is inconsistent.