Graph The System Below And Write Its Solution

The moment you need to graph the system below and write its solution, the process involves plotting each equation on the coordinate plane and identifying where they intersect. This visual approach not only clarifies the relationship between the equations but also provides an intuitive way to determine the ordered pair(s) that satisfy every equation in the system simultaneously. Whether you are solving a simple pair of linear equations or tackling a more complex set involving inequalities, mastering these steps will give you a solid foundation for tackling algebraic problems in school, business, or engineering contexts.

Understanding the Coordinate Plane

The coordinate plane is a two‑dimensional surface formed by two perpendicular number lines: the horizontal x‑axis and the vertical y‑axis. Worth adding: their intersection point is called the origin (0, 0). Any point on this plane can be described by an ordered pair (x, y), where x indicates the horizontal distance from the origin and y indicates the vertical distance Simple, but easy to overlook..

When graphing a linear equation, you typically convert it to slope‑intercept form, written as

[ y = mx + b ]

Here, m represents the slope (the rate of change or steepness of the line) and b is the y‑intercept (the point where the line crosses the y‑axis). Recognizing this form makes it easy to plot the line quickly and accurately Practical, not theoretical..

Converting to Slope‑Intercept Form

Before you can plot, ensure each equation is in slope‑intercept form. Here's one way to look at it: consider the system

1. (2x - y = -1)
2. (x + y = 5)

To rewrite them:

1. (2x - y = -1 ;\Rightarrow; -y = -2x - 1 ;\Rightarrow; y = 2x + 1)
2. (x + y = 5 ;\Rightarrow; y = -x + 5)

Now both equations are ready for graphing:

• Equation 1: (y = 2x + 1) → slope m = 2, y‑intercept b = 1
• Equation 2: (y = -x + 5) → slope m = ‑1, y‑intercept b = 5

Plotting the Lines

Step‑by‑step plotting guide

1. Mark the y‑intercept for each line.

   • For (y = 2x + 1), plot the point (0, 1).
   • For (y = -x + 5), plot the point (0, 5).

2. Use the slope to find a second point.

   • Slope = rise/run.
   • For (y = 2x + 1) (slope = 2 = 2/1): from (0, 1) move up 2 and right 1 to reach (1, 3).
   • For (y = -x + 5) (slope = ‑1 = ‑1/1): from (0, 5) move down 1 and right 1 to reach (1, 4).

3. Draw a straight line through each pair of points. Extend the lines across the graph; they represent all possible solutions for each individual equation.

Finding the Intersection (Solution)

The solution to a system of linear equations is the ordered pair where the lines intersect. There are three possible outcomes:

1. One unique solution – lines intersect at a single point.
2. No solution – lines are parallel (same slope, different y‑intercepts).
3. Infinitely many solutions – lines are identical (same slope and y‑intercept).

For the given system, the lines are not parallel, so they intersect at exactly one point. To find this point algebraically, set the two expressions for y equal:

[ 2x + 1 = -x + 5 ]

Solve for x:

[ 2x + x = 5 - 1 \ 3x = 4 \ x = \frac{4}{3} ]

Now substitute x back into either equation to find y:

[ y = 2\left(\frac{4}{3}\right) + 1 = \frac{8}{3} + 1 = \frac{8}{3} + \frac{3}{3} = \frac{11}{3} ]

Thus, the intersection point is (\left(\frac{4}{3}, \frac{11}{3}\right)). This ordered pair is the solution to the system, meaning it satisfies both original equations Worth keeping that in mind..

Graphing Inequalities (Optional Extension)

If the system includes inequalities (e.g., (y > 2x + 1) and (y \le -x + 5)), the graphing process adds shading:

1. Graph the boundary line as if it were an equation. Use a dashed line for strict inequalities (>, <) and a solid line for inclusive inequalities (≤, ≥).
2. Choose a test point not on the line (often (0, 0)). Plug it into the inequality.
3. Shade the region that makes the inequality true. The overlapping shaded area represents the solution set for the system of inequalities.

Tips for Accuracy

• Label axes clearly with numbers to avoid misreading coordinates.
• Use a ruler to draw straight lines; this helps maintain consistency.
• Double‑check calculations for slope and intercept; a small arithmetic error shifts the entire line.
• Verify the intersection by substituting the ordered pair back into both original equations.

Common Mistakes to Avoid

• Mixing up slope sign: Remember that a positive slope rises left‑to‑right, while a negative slope falls.
• Forgetting to flip the inequality sign when multiplying or dividing by a negative number (especially important