To solve the system of equations graphically, let’s first identify the two equations we will be working with:
- Blue Line: \( y = \frac{2}{3}x - 2 \)
- Red Line: \( y = -\frac{1}{2}x + 5 \)
To find the solution of the system, we need to determine the point where these two lines intersect.
Step 1: Find the Intersection Point
To find the point of intersection, set the two equations equal to each other:
\[ \frac{2}{3}x - 2 = -\frac{1}{2}x + 5 \]
Step 2: Solve for \( x \)
Multiply every term by 6 to eliminate the fractions:
\[ 6 \left(\frac{2}{3}x\right) - 6(2) = 6\left(-\frac{1}{2}x\right) + 6(5) \]
This simplifies to:
\[ 4x - 12 = -3x + 30 \]
Now, combine like terms:
\[ 4x + 3x = 30 + 12 \]
\[ 7x = 42 \]
Now, divide by 7:
\[ x = 6 \]
Step 3: Substitute \( x \) Back to Find \( y \)
Now, we can substitute \( x = 6 \) back into either equation to find \( y \). Let’s use the blue line equation:
\[ y = \frac{2}{3}(6) - 2 \]
Calculating this gives:
\[ y = 4 - 2 = 2 \]
Solution
Thus, the solution to the system of equations, or the point of intersection, is:
(6, 2)
You can then enter the solution as:
Enter the solution: \( (6, 2) \)
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