How can we solve the system of equations given by 2x + y = 3 and 4x + 2y = 2 graphically?

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Solving the System of Equations Graphically

To solve the system of equations defined by:

  • Equation 1: 2x + y = 3
  • Equation 2: 4x + 2y = 2

we will follow these steps:

Step 1: Rewrite the Equations in Slope-Intercept Form

First, we need to express both equations in the slope-intercept form (y = mx + b), where m is the slope and b is the y-intercept.

For Equation 1 (2x + y = 3):

  • Isolate y: y = 3 – 2x

For Equation 2 (4x + 2y = 2):

  • Isolate y: 2y = 2 – 4x
  • Then, y = 1 – 2x

Step 2: Graph the Equations

Now that we have the equations in the slope-intercept form, we can graph both lines on the same axis:

  1. Graph Equation 1: Start by plotting the y-intercept (0, 3). Next, use the slope (-2) to find another point. From (0, 3), go down 2 units and to the right 1 unit to point (1, 1). Connect these points to form the line.
  2. Graph Equation 2: Begin plotting from the y-intercept (0, 1). Use the slope (-2) as well; from (0, 1), go down 2 units and to the right 1 unit to point (1, -1). Connect these points to outline this second line.

Step 3: Identify the Intersection Point

The solution to the system of equations is the point where both lines intersect. Look at the graph to find the coordinates of the intersection point. This point represents the values of x and y that satisfy both equations simultaneously.

Finding the Intersection:

By analyzing the lines, we can see that these lines intersect at the point (1, 1).

Final Answer

Thus, the solution to the system of equations is:

  • x = 1
  • y = 1

This means that the coordinate point (1, 1) solves both equations.

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