Solve the following system of linear equations graphically: x + 2y = 3, 2x - 3y + 8 = 0
Understand the Problem
The question is asking us to solve a system of linear equations by graphing them. This involves finding the point where the two lines represented by the equations intersect.
Answer
The intersection point is \((2, 1)\).
Answer for screen readers
The solution to the system of equations is the intersection point, which is ((2, 1)).
Steps to Solve
- Rewrite the equations in slope-intercept form
For the first equation, $x + 2y = 3$: Subtract $x$ from both sides: $$2y = -x + 3$$ Now divide by $2$: $$y = -\frac{1}{2}x + \frac{3}{2}$$
For the second equation, $2x - 3y + 8 = 0$: Rearranging gives: $$-3y = -2x - 8$$ Dividing by $-3$: $$y = \frac{2}{3}x + \frac{8}{3}$$
- Graph the equations
Plot the two equations on a graph.
- For $y = -\frac{1}{2}x + \frac{3}{2}$, find points by substituting values for $x$.
- For $y = \frac{2}{3}x + \frac{8}{3}$, also substitute values for $x$ to find points.
- Find the intersection point
Look for the point where the two lines intersect on the graph. This point is the solution to the system of equations.
The solution to the system of equations is the intersection point, which is ((2, 1)).
More Information
The intersection point represents the values of (x) and (y) that satisfy both equations simultaneously. This method of solving linear equations graphically helps visualize the relationship between the equations.
Tips
- Misplotting: Ensure that you plot points accurately on the graph.
- Incorrect line drawing: Draw straight lines through the plotted points; error-prone sketches can lead to an incorrect intersection point.
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