To find the approximate solution to the system of equations given by:
- \(y = 0.5x + 3.5\)
- \(y = -\frac{2}{3}x + \frac{1}{3}\)
we need to find the point where these two lines intersect.
Step 1: Set the equations equal to each other to find \(x\):
\[ 0.5x + 3.5 = -\frac{2}{3}x + \frac{1}{3} \]
Step 2: To eliminate the fractions, let's multiply through by 6, the least common multiple of the denominators:
\[ 6(0.5x) + 6(3.5) = 6\left(-\frac{2}{3}x\right) + 6\left(\frac{1}{3}\right) \]
This simplifies to:
\[ 3x + 21 = -4x + 2 \]
Step 3: Now, combine the like terms:
\[ 3x + 4x = 2 - 21 \]
\[ 7x = -19 \]
\[ x = -\frac{19}{7} \approx -2.71 \]
Step 4: Substitute \(x\) back into one of the original equations to find \(y\). We'll use the first equation:
\[ y = 0.5(-\frac{19}{7}) + 3.5 \]
Calculate \(0.5 \times -\frac{19}{7}\):
\[ y = -\frac{19}{14} + \frac{49}{14} = \frac{30}{14} = \frac{15}{7} \approx 2.14 \]
So, the approximate intersection point is:
\[ \left(-2.71, 2.14\right) \]
Given the options:
\((-2.7, 2.1)\)
\((-2.1, 2.7)\)
\((2.1, 2.7)\)
\((2.7, 2.1)\)
The closest match is:
\(\boxed{(-2.7, 2.1)}\)
1 answer