Question

here is a graph. write an equation that can represent each line. equation 1 equation 2 estimate the solution to the system.

Explanation:

Step1: Analyze the purple line (Equation 1)

The purple line passes through (0, 2) and has a slope. Let's calculate the slope. From (0,2) to (1,5), the slope $m = \frac{5 - 2}{1 - 0} = 3$. Using slope - intercept form $y = mx + b$, where $b = 2$ (y - intercept), so the equation is $y = 3x + 2$.

Step2: Analyze the red line (Equation 2)

The red line passes through (0, 4) and (4, 2). The slope $m=\frac{2 - 4}{4 - 0}=\frac{- 2}{4}=-\frac{1}{2}$. Using slope - intercept form $y=mx + b$, with $b = 4$, the equation is $y=-\frac{1}{2}x + 4$.

Step3: Find the intersection (solution)

Set $3x + 2=-\frac{1}{2}x + 4$. Add $\frac{1}{2}x$ to both sides: $3x+\frac{1}{2}x+2 = 4$. Combine like terms: $\frac{6x + x}{2}+2 = 4\Rightarrow\frac{7x}{2}=2$. Multiply both sides by $\frac{2}{7}$: $x=\frac{4}{7}\approx0.57$. Then $y = 3\times\frac{4}{7}+2=\frac{12 + 14}{7}=\frac{26}{7}\approx3.71$. But by looking at the graph, the intersection seems to be around (1, 5)? Wait, no, let's re - check. Wait, maybe my initial points for purple line are wrong. Let's take another approach. Looking at the purple line: when $x = 0$, $y = 2$; when $x = 1$, $y = 5$ (so slope 3). Red line: when $x = 0$, $y = 4$; when $x = 2$, $y = 3$ (slope - 0.5). Setting $3x + 2=-\frac{1}{2}x + 4$. Multiply both sides by 2: $6x+4=-x + 8$. Add $x$ to both sides: $7x+4 = 8$. Subtract 4: $7x = 4$, $x=\frac{4}{7}\approx0.57$, $y=3\times\frac{4}{7}+2=\frac{12 + 14}{7}=\frac{26}{7}\approx3.71$. But maybe from the graph, the intersection is at (1, 5)? Wait, no, maybe I made a mistake in the red line's y - intercept. Wait, looking at the graph, the red line passes through (0,4) and (8,0). So slope is $\frac{0 - 4}{8 - 0}=-\frac{1}{2}$. So equation $y=-\frac{1}{2}x + 4$. The purple line: when $x = 0$, $y = 2$; when $x = 2$, $y = 8$? No, wait the purple line goes from (- 6, - 16) to (4, 14)? Wait, maybe the purple line has a slope of 3: from (0,2) to (2,8), slope is 3. Then red line: from (0,4) to (8,0), slope - 0.5. Solving $3x + 2=-\frac{1}{2}x + 4$: $3x+\frac{1}{2}x=4 - 2$, $\frac{7x}{2}=2$, $x=\frac{4}{7}\approx0.57$, $y = 3\times\frac{4}{7}+2\approx3.71$. But visually, the intersection is around (1, 5)? Wait, maybe my initial assumption of the purple line's y - intercept is wrong. Let's look at the grid again. The purple line crosses the y - axis at (0,2)? Wait, no, in the graph, the purple line at x = 0 is at y = 2? Wait, the grid has y - axis from - 5 to 10, x - axis from - 5 to 10. Let's take two points on purple line: (0, 2) and (1, 5) (so slope 3). Red line: (0, 4) and (4, 2) (slope - 0.5). The intersection: solve $3x + 2=-\frac{1}{2}x + 4$. $3x+\frac{1}{2}x=4 - 2$, $\frac{7x}{2}=2$, $x=\frac{4}{7}\approx0.57$, $y=\frac{26}{7}\approx3.71$. But maybe the problem expects approximate values from the graph. Looking at the graph, the two lines intersect at a point where x is around 1 and y is around 5? Wait, no, maybe I messed up the lines. Let's re - identify:

Wait, the purple line: when x = 0, y = 2; when x = 2, y = 8 (so slope 3). Red line: when x = 0, y = 4; when x = 8, y = 0 (slope - 0.5). So equation of purple: $y = 3x+2$, red: $y=-\frac{1}{2}x + 4$.

To find intersection:

$3x + 2=-\frac{1}{2}x + 4$

$3x+\frac{1}{2}x=4 - 2$

$\frac{6x + x}{2}=2$

$\frac{7x}{2}=2$

$x=\frac{4}{7}\approx0.57$

$y=3\times\frac{4}{7}+2=\frac{12 + 14}{7}=\frac{26}{7}\approx3.71$

But maybe the graph is drawn with approximate points. Alternatively, if we look at the graph, the purple line and red line intersect at (1, 5)? No, when x = 1, purple line: $y = 3(1)+2 = 5$, red line: $y=-\frac{1}{2}(1)+4 = 3.5$. Not the same. Wait, maybe the purple line's y - intercept is 2 and red line's y - intercept is 4, and the intersection is at x = 0.8, y = 4.4? But maybe the problem is designed to have integer solutions. Wait, maybe I made a mistake in the slope of the purple line. Let's take (0, 2) and (2, 8): slope is 3. Red line: (0, 4) and (4, 2): slope - 0.5.

Answer:

Equation 1: $y = 3x + 2$

Equation 2: $y=-\frac{1}{2}x + 4$

Solution (intersection): $(\frac{4}{7},\frac{26}{7})\approx(0.57, 3.71)$ (or approximate from graph as (1, 5) if there was a drawing error in my analysis)