Question

for exercises 10-12, write an equation that matches each graph.
10.
11.

Explanation:

Exercise 10

Step1: Identify two points

From the graph, let's take two points. Let's say \((1, 2)\) and \((2, 1)\) (we can check the grid, but actually, looking at the slope, let's find slope first. Wait, better to take clear points. Let's see, when \(x = 1\), \(y = 2\)? Wait, no, let's check the intercept. Wait, maybe better to use the slope formula. Let's take two points: say \((-2, 4)\) and \((-1, 3)\)? Wait, no, maybe the line passes through \((1, 2)\) and \((2, 1)\)? Wait, actually, let's look at the grid. Let's assume each grid is 1 unit. Let's take two points: \((0, 2)\)? Wait, no, maybe the line has a slope of \(-1\)? Wait, let's check the rise over run. From a point to another, if we move right 1, down 1, so slope \(m=-1\). Now, find the y-intercept. Let's see where it crosses the y-axis. Looking at the graph, when \(x = 0\), \(y = 2\)? Wait, no, maybe I made a mistake. Wait, let's take two clear points. Let's say \((1, 2)\) and \((3, 0)\). Then slope \(m=\frac{0 - 2}{3 - 1}=\frac{-2}{2}=-1\). Now, using point-slope form \(y - y_1=m(x - x_1)\), using \((3, 0)\): \(y - 0=-1(x - 3)\), so \(y=-x + 3\). Wait, let's check with another point. If \(x = 1\), \(y=-1 + 3=2\), which matches. If \(x = 2\), \(y=-2 + 3=1\), which also matches. So the equation is \(y=-x + 3\).

Step2: Verify

Check if the line passes through the points. For \(x = 1\), \(y = 2\): \(2=-1 + 3=2\), correct. For \(x = 3\), \(y = 0\): \(0=-3 + 3=0\), correct. So the equation \(y=-x + 3\) matches the graph.

Step1: Find slope and intercept

First, find two points on the line. Let's take \((0, -1)\) (y-intercept) and \((-2, 1)\). The slope \(m=\frac{1 - (-1)}{-2 - 0}=\frac{2}{-2}=-1\). The y-intercept \(b=-1\) (since it crosses the y-axis at \((0, -1)\)). So using slope-intercept form \(y=mx + b\), we get \(y=-x - 1\). Wait, let's check another point. Take \((-1, 0)\): \(y=-(-1)-1=1 - 1=0\), which matches. Take \((1, -2)\): \(y=-1 - 1=-2\), which also matches.

Step2: Confirm

Check the slope: from \((-2, 1)\) to \((0, -1)\), run is 2, rise is -2, so slope is \(-1\), correct. Y-intercept at \((0, -1)\), correct. So the equation is \(y=-x - 1\).

Answer:

\(y = -x + 3\)

Exercise 11