To solve the given system of equations graphically, we need to plot both equations on a coordinate plane and find the point where the two lines intersect. The given equations are:
1. [tex]\( y = \frac{1}{2} x - 5 \)[/tex]
2. [tex]\( y = -\frac{1}{3} x \)[/tex]
Step-by-Step Solution:
1. First Equation: [tex]\( y = \frac{1}{2} x - 5 \)[/tex]
- This is a linear equation in slope-intercept form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
- The slope [tex]\( m \)[/tex] is [tex]\( \frac{1}{2} \)[/tex], and the y-intercept [tex]\( b \)[/tex] is [tex]\(-5\)[/tex].
- To plot this equation, start at the y-intercept [tex]\((0, -5)\)[/tex].
- From [tex]\((0, -5)\)[/tex], use the slope [tex]\( \frac{1}{2} \)[/tex] (which means that for every 2 units along the x-axis, the line rises 1 unit along the y-axis).
- Another point on the line can be found by moving 2 units to the right (positive x direction) and 1 unit up (positive y direction). This gives the point (2, -4).
2. Second Equation: [tex]\( y = -\frac{1}{3} x \)[/tex]
- This equation is also in slope-intercept form [tex]\( y = mx + b \)[/tex], where the slope [tex]\( m \)[/tex] is [tex]\(-\frac{1}{3} \)[/tex] and the y-intercept [tex]\( b \)[/tex] is [tex]\(0\)[/tex].
- To plot this equation, start at the origin [tex]\((0, 0)\)[/tex].
- From [tex]\((0, 0)\)[/tex], use the slope [tex]\(-\frac{1}{3} \)[/tex] (which means that for every 3 units along the x-axis, the line falls 1 unit along the y-axis).
- Another point on the line can be found by moving 3 units to the right (positive x direction) and 1 unit down (negative y direction). This gives the point (3, -1).
3. Graph the Equations:
- Graph the first equation by plotting the points [tex]\((0, -5)\)[/tex] and [tex]\((2, -4)\)[/tex], then draw the line through these points.
- Graph the second equation by plotting the points [tex]\((0, 0)\)[/tex] and [tex]\((3, -1)\)[/tex], then draw the line through these points.
4. Find the Intersection:
- Observe the graph to find the point where the two lines intersect. The correct x and y coordinates of this intersection point are [tex]\(x = 6\)[/tex] and [tex]\(y = -2\)[/tex].
Hence, the solution to the system of equations is the point (6, -2).
So, the correct answer is:
[tex]\(\{ (6, -2) \}\)[/tex]
