Question

which is the approximate solution to the system $y = 0.5x + 3.5$ and $y = -\frac{2}{3}x + \frac{1}{3}$ shown on the graph?
$(-2.1, 2.7)$
$(2.7, 2.1)$
$(2.1, 2.7)$
$(-2.7, 2.1)$

Explanation:

Step1: Recall solution of system

The solution to a system of linear equations \(y = m_1x + b_1\) and \(y = m_2x + b_2\) is the point of intersection of their graphs. So we can solve the system by setting the two equations equal: \(0.5x + 3.5=-\frac{2}{3}x+\frac{1}{3}\).

Step2: Solve for \(x\)

First, convert \(0.5=\frac{1}{2}\). So the equation becomes \(\frac{1}{2}x + 3.5=-\frac{2}{3}x+\frac{1}{3}\). Multiply all terms by 6 to eliminate denominators: \(6\times\frac{1}{2}x+6\times3.5 = 6\times(-\frac{2}{3}x)+6\times\frac{1}{3}\). This simplifies to \(3x + 21=- 4x + 2\). Add \(4x\) to both sides: \(3x + 4x+21=-4x + 4x+2\) \(\Rightarrow 7x+21 = 2\). Subtract 21 from both sides: \(7x=2 - 21=-19\). Then \(x=\frac{-19}{7}\approx - 2.714\approx - 2.7\) (wait, no, wait, maybe I made a mistake. Wait, let's check the graph. The blue line \(y = 0.5x + 3.5\) has a positive slope, red line \(y=-\frac{2}{3}x+\frac{1}{3}\) has negative slope. The intersection point: let's check the x - coordinate. From the graph, the intersection is at x negative? Wait, no, wait the options: one is (-2.1,2.7), (-2.7,2.1), etc. Wait, maybe better to plug x into one of the equations. Let's take the first equation \(y = 0.5x+3.5\). Let's check option D: (-2.7,2.1). Plug x=-2.7: \(y=0.5\times(-2.7)+3.5=-1.35 + 3.5 = 2.15\approx2.1\). Let's check option A: (-2.1,2.7). \(y=0.5\times(-2.1)+3.5=-1.05 + 3.5 = 2.45\approx2.5
eq2.7\). Option B: (2.7,2.1): \(y=0.5\times2.7 + 3.5 = 1.35+3.5 = 4.85
eq2.1\). Option C: (2.1,2.7): \(y=0.5\times2.1+3.5 = 1.05 + 3.5 = 4.55
eq2.7\). Wait, but when we solved the equation: \(0.5x+3.5=-\frac{2}{3}x+\frac{1}{3}\). Let's re - solve:

\(\frac{1}{2}x+\frac{7}{2}=-\frac{2}{3}x+\frac{1}{3}\)

Multiply by 6: \(3x + 21=-4x + 2\)

\(3x + 4x=2 - 21\)

\(7x=-19\)

\(x =-\frac{19}{7}\approx - 2.714\approx - 2.7\)

Then \(y=0.5\times(-\frac{19}{7})+\frac{7}{2}=\frac{-19}{14}+\frac{49}{14}=\frac{30}{14}=\frac{15}{7}\approx2.14\approx2.1\)

So the solution is \((-2.7,2.1)\), which is option D. Wait, but let's check the graph again. The blue line: when x = 0, y = 3.5. The red line: when x = 0, y=\(\frac{1}{3}\approx0.333\). The intersection: x is negative (since red line at x = 0 is low, blue line at x = 0 is high, so they intersect at x negative). So x is negative, y is positive. So among the options, (-2.7,2.1) is the one that fits. Wait, but earlier when I miscalculated, but now with the equation, x≈-2.7, y≈2.1. So the correct option is D: \((-2.7,2.1)\)

Answer:

\(\boldsymbol{(-2.7, 2.1)}\) (corresponding to the option with identifier, e.g., if it's option D: D. \((-2.7, 2.1)\))