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A system of equations is shown.

y = 5x - 3
y = -2x + 9

PART A

Graph the system of equations.

1 answer

To graph the system of equations given by:

  1. \( y = 5x - 3 \)
  2. \( y = -2x + 9 \)

you can follow these steps:

Graphing the first equation: \( y = 5x - 3 \)

  1. Identify the y-intercept: The y-intercept is \(-3\). This means the point \( (0, -3) \) is on the graph.
  2. Identify the slope: The slope is \(5\). This means for every 1 unit you move to the right (positive \(x\)), you move 5 units up (positive \(y\)).
  3. Plot another point: Starting from the y-intercept (0, -3), move right 1 unit to \(x = 1\) and up 5 units to \(y = 2\). This gives the point \( (1, 2) \).
  4. Draw the line: Draw a straight line through the points \( (0, -3) \) and \( (1, 2) \).

Graphing the second equation: \( y = -2x + 9 \)

  1. Identify the y-intercept: The y-intercept is \(9\). This means the point \( (0, 9) \) is on the graph.
  2. Identify the slope: The slope is \(-2\). This means for every 1 unit you move to the right (positive \(x\)), you move 2 units down (negative \(y\)).
  3. Plot another point: Starting from the y-intercept (0, 9), move right 1 unit to \(x = 1\) and down 2 units to \(y = 7\). This gives the point \( (1, 7) \).
  4. Draw the line: Draw a straight line through the points \( (0, 9) \) and \( (1, 7) \).

Finding the intersection

The solution to the system of equations, or the point where the two lines intersect, can be found graphically by determining the point at which the lines cross.

Final Step: Plotting the graph

You can sketch the two lines on the same graph, ensuring you label them appropriately. The intersection point will give you the solution to the system of equations. If you are using graphing software or a graphing calculator, you may be able to find the intersection point numerically as well.

Note: Example Point

For a quick check on where these lines intersect mathematically:

Solve for \(x\) and \(y\) by setting the equations equal: \[ 5x - 3 = -2x + 9 \] Combine like terms and solve for \(x\). You can substitute back to find \(y\).

Conclusion

Use graph paper and plot the points accurately to visualize the system of equations!