Certainly! To solve this system of equations and graphically determine their intersection point, follow these detailed steps:
### Step-by-Step Graphical Solution
1. Rewrite the Equations for Clarity:
- The first equation is [tex]\( y = \frac{1}{4}x + 3 \)[/tex]
- The second equation is [tex]\( y = 2x + 10 \)[/tex]
2. Identify and Plot Key Points for Each Equation:
- For the first equation [tex]\( y = \frac{1}{4} x + 3 \)[/tex]:
- Identify the y-intercept (where [tex]\( x = 0 \)[/tex]): [tex]\( y = 3 \)[/tex].
- Choose another value for [tex]\( x \)[/tex] to find another point. For instance, when [tex]\( x = 4 \)[/tex], [tex]\( y = \frac{1}{4}(4) + 3 = 1 + 3 = 4 \)[/tex].
Thus, the points are [tex]\( (0, 3) \)[/tex] and [tex]\( (4, 4) \)[/tex].
- For the second equation [tex]\( y = 2x + 10 \)[/tex]:
- Identify the y-intercept (where [tex]\( x = 0 \)[/tex]): [tex]\( y = 10 \)[/tex].
- Choose another value for [tex]\( x \)[/tex] to find another point. For instance, when [tex]\( x = -2 \)[/tex], [tex]\( y = 2(-2) + 10 = -4 + 10 = 6 \)[/tex].
Thus, the points are [tex]\( (0, 10) \)[/tex] and [tex]\( (-2, 6) \)[/tex].
3. Draw Each Line on the Graph Paper:
- Plot the points [tex]\( (0, 3) \)[/tex] and [tex]\( (4, 4) \)[/tex] for the first equation. Draw a straight line through these points.
- Plot the points [tex]\( (0, 10) \)[/tex] and [tex]\( (-2, 6) \)[/tex] for the second equation. Draw a straight line through these points.
4. Find the Intersection Point:
- Observe the point where the two lines intersect on the graph.
According to the numerical solution provided earlier, the intersection point of these lines is:
- [tex]\( x = -4 \)[/tex]
- [tex]\( y = 2 \)[/tex]
Thus, the solution to the system of equations is:
[tex]\[
( x, y ) = ( -4, 2 )
\][/tex]
So, you would enter:
- x = -4
- y = 2
This intersection point is the solution to the system of the given equations.
