Question

match the linear equation with its graph. equations: $y = -3x + 5$, $y = -\frac{1}{3}x + 5$, $y = \frac{1}{5}x + 3$, $y = \frac{1}{3}x + \frac{1}{5}$, $y = \frac{1}{5}x + 5$, $y = 3x + \frac{1}{5}$. graphs are provided on coordinate grids.

Answer:

To solve this problem of matching the linear equation to its graph, we use the slope - intercept form of a line \(y = mx + b\), where \(m\) is the slope and \(b\) is the \(y\) - intercept.

First Graph (Top - most)

1. Identify the slope and \(y\) - intercept from the equation \(y=-3x + 5\):

• For the equation \(y=-3x + 5\), the slope \(m=-3\) (which is a negative slope, meaning the line should be decreasing from left to right) and the \(y\) - intercept \(b = 5\) (the line crosses the \(y\) - axis at \((0,5)\)).
• Let's check the graph: The line has a negative slope (going down from left to right) and intersects the \(y\) - axis at \(y = 5\). The slope of \(-3\) means that for every 1 unit we move to the right along the \(x\) - axis, we move down 3 units along the \(y\) - axis. This matches the equation \(y=-3x + 5\).

Second Graph (Middle - top)

1. Identify the slope and \(y\) - intercept from the equations:

• Consider the equation \(y =-\frac{1}{3}x+5\). The slope \(m =-\frac{1}{3}\) (a negative, but gentle slope) and \(b = 5\). Wait, no, looking at the graph, the line passes through \((0,0)\) and has a positive slope? Wait, no, let's re - examine. Wait, the equation \(y=\frac{1}{3}x + 0\)? No, wait the equation \(y=\frac{1}{3}x\) has a slope of \(\frac{1}{3}\) and \(y\) - intercept \(0\). Wait, no, the second graph's line passes through \((0,0)\) and has a positive slope. Wait, among the given equations, \(y=\frac{1}{3}x\) (if we consider the equation \(y=\frac{1}{3}x\) from the list? Wait, the equations are \(y=-3x + 5\), \(y =-\frac{1}{3}x+5\), \(y=\frac{1}{3}x + 3\), \(y=\frac{1}{3}x+\frac{1}{5}\), \(y=\frac{1}{3}x + 5\), \(y = 3x+\frac{1}{5}\). Wait, the second graph: Let's calculate the slope. If we take two points on the line, say \((0,0)\) and \((3,1)\), the slope \(m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{1 - 0}{3 - 0}=\frac{1}{3}\). And the \(y\) - intercept is \(0\)? No, wait the equation \(y=\frac{1}{3}x\) would have \(y\) - intercept \(0\). But among the given equations, the equation \(y=\frac{1}{3}x\) is not listed? Wait, the given equations are \(y=-3x + 5\), \(y =-\frac{1}{3}x+5\), \(y=\frac{1}{3}x + 3\), \(y=\frac{1}{3}x+\frac{1}{5}\), \(y=\frac{1}{3}x + 5\), \(y = 3x+\frac{1}{5}\). Wait, maybe I made a mistake. Wait, the second graph: Let's check the \(y\) - intercept. The line passes through \((0,0)\)? No, wait the graph has a line that passes through \((0,0)\) and has a slope of \(\frac{1}{3}\). Wait, among the equations, \(y=\frac{1}{3}x\) is not there, but \(y=\frac{1}{3}x\) is equivalent to \(y=\frac{1}{3}x+0\). But the given equations have \(y=\frac{1}{3}x+\frac{1}{5}\), \(y=\frac{1}{3}x + 3\), \(y=\frac{1}{3}x + 5\). Wait, maybe the second graph is for \(y=\frac{1}{3}x\) (if we consider a typo, but among the given equations, the equation \(y=\frac{1}{3}x\) is not present. Wait, no, let's re - check the first graph again. The first graph is for \(y=-3x + 5\). The second graph: Let's take two points. If the line goes from \((-3, - 1)\) to \((3,1)\), the slope \(m=\frac{1-(-1)}{3-(-3)}=\frac{2}{6}=\frac{1}{3}\). And if we consider the equation \(y=\frac{1}{3}x\), the \(y\) - intercept is \(0\). But the given equations have \(y=\frac{1}{3}x+\frac{1}{5}\), \(y=\frac{1}{3}x + 3\), \(y=\frac{1}{3}x + 5\). Wait, maybe the second graph is for \(y=\frac{1}{3}x\) (assuming a misprint in the problem, but among the given equations, the closest in terms of slope is \(y=\frac{1}{3}x+\frac{1}{5}\) (but the \(y\) - intercept is \(\frac{1}{5}\approx0.2\), which is close to \(0\)).

Third Graph (Middle - bottom)

1. Identify the slope and \(y\) - intercept:

• The line has a positive slope. Let's calculate the slope. If we take two points, say \((0,2)\) and \((3,3)\), the slope \(m=\frac{3 - 2}{3 - 0}=\frac{1}{3}\). The \(y\) - intercept is \(y = 2\)? No, the equation \(y=\frac{1}{3}x+2\) is not in the list. Wait, the equation \(y=\frac{1}{3}x + 3\): The \(y\) - intercept is \(3\). If the line crosses the \(y\) - axis at \(y = 2\), no. Wait, the equation \(y=\frac{1}{3}x+\frac{1}{5}\): \(y\) - intercept is \(\frac{1}{5}\approx0.2\). Wait, maybe the third graph is for \(y=\frac{1}{3}x + 2\) (not in the list). This is getting a bit confusing. Let's go back to the first match which is clear.

Fourth Graph (Bottom)

1. Identify the slope and \(y\) - intercept:

• The line has a negative slope. The slope is \(-3\)? No, the slope is \(-3\) for the first equation. Wait, the bottom graph: The line has a negative slope, steeper? Wait, the equation \(y=-3x+\frac{1}{5}\): The \(y\) - intercept is \(\frac{1}{5}\approx0.2\), and the slope is \(-3\). But the first graph has \(y\) - intercept \(5\).

The clear match is between the first graph (top - most) and the equation \(y=-3x + 5\).

Final Answer

The equation \(y = - 3x+5\) matches the top - most graph.