Question

a) write an equation for the total cost of taking students and teachers to the state...
b) write the equation from part a in slope - intercept form.
unit 2 - item 10
create a story that models the graph below.
graph with distance from home on y - axis and time on x - axis
unit 2 - item 11
jayleen and ja’keria were in study skills class discussing functions and linear relationships. jayleen gave ja’keria the steps below to complete. can you help ja’keria model and represent the given statements?
a) the x - intercept of a linear equation is 3 and the y - intercept is 7. graph this line on the coordinate plane.
coordinate plane graph
confidential and secure - not for public distribution
this work is licensed under a creative commons attribution - noncommercial - sharealike 4.0 international license
georgia department of education
page 16 of 46
may 2024

Explanation:

To solve part A of Unit 2 - Item 11 (graphing the line with x - intercept 3 and y - intercept 7), we can follow these steps:

Step 1: Recall the intercept form of a linear equation

The intercept form of a linear equation is given by $\frac{x}{a}+\frac{y}{b} = 1$, where $a$ is the x - intercept and $b$ is the y - intercept.
Given that the x - intercept $a = 3$ and the y - intercept $b=7$, the equation of the line in intercept form is $\frac{x}{3}+\frac{y}{7}=1$.

Step 2: Find two points on the line

• The x - intercept is the point where the line crosses the x - axis. At the x - axis, $y = 0$. So when $y = 0$, from the equation $\frac{x}{3}+\frac{0}{7}=1$, we get $\frac{x}{3}=1$, and $x = 3$. So one point on the line is $(3,0)$.
• The y - intercept is the point where the line crosses the y - axis. At the y - axis, $x = 0$. So when $x = 0$, from the equation $\frac{0}{3}+\frac{y}{7}=1$, we get $\frac{y}{7}=1$, and $y = 7$. So another point on the line is $(0,7)$.

Step 3: Plot the points and draw the line

• On the coordinate plane, find the point $(3,0)$ (3 units to the right of the origin on the x - axis) and the point $(0,7)$ (7 units up from the origin on the y - axis).
• Use a straight - edge to draw a line passing through these two points.

If we want to write the equation in slope - intercept form ($y=mx + c$, where $m$ is the slope and $c$ is the y - intercept) first:

Step 1: Calculate the slope

The slope $m$ of a line passing through two points $(x_1,y_1)$ and $(x_2,y_2)$ is given by $m=\frac{y_2 - y_1}{x_2 - x_1}$.
We have the points $(3,0)$ and $(0,7)$. Let $(x_1,y_1)=(3,0)$ and $(x_2,y_2)=(0,7)$. Then $m=\frac{7 - 0}{0 - 3}=\frac{7}{-3}=-\frac{7}{3}$.
The y - intercept $c = 7$ (from the given y - intercept). So the equation of the line in slope - intercept form is $y=-\frac{7}{3}x + 7$.

To graph the line using the slope - intercept form:

• Start by plotting the y - intercept $(0,7)$.
• The slope $m =-\frac{7}{3}$ means that for every 3 units we move to the right (increase in $x$ by 3), we move down 7 units (decrease in $y$ by 7). From the point $(0,7)$, if we move 3 units to the right (to $x = 3$) and 7 units down (to $y=7 - 7=0$), we reach the point $(3,0)$, which is the x - intercept. Then draw a line through $(0,7)$ and $(3,0)$.

To answer the question of graphing the line:

1. Plot the points $(3,0)$ (x - intercept) and $(0,7)$ (y - intercept) on the coordinate plane.
2. Draw a straight line passing through these two points.

If we were to describe the graph: The line intersects the x - axis at $(3,0)$ and the y - axis at $(0,7)$. It has a negative slope of $-\frac{7}{3}$, so it is a line that goes from the second quadrant (near the y - axis) to the fourth quadrant (near the x - axis) as $x$ increases.

Answer:

To solve part A of Unit 2 - Item 11 (graphing the line with x - intercept 3 and y - intercept 7), we can follow these steps:

Step 1: Recall the intercept form of a linear equation

The intercept form of a linear equation is given by $\frac{x}{a}+\frac{y}{b} = 1$, where $a$ is the x - intercept and $b$ is the y - intercept.
Given that the x - intercept $a = 3$ and the y - intercept $b=7$, the equation of the line in intercept form is $\frac{x}{3}+\frac{y}{7}=1$.

Step 2: Find two points on the line

• The x - intercept is the point where the line crosses the x - axis. At the x - axis, $y = 0$. So when $y = 0$, from the equation $\frac{x}{3}+\frac{0}{7}=1$, we get $\frac{x}{3}=1$, and $x = 3$. So one point on the line is $(3,0)$.
• The y - intercept is the point where the line crosses the y - axis. At the y - axis, $x = 0$. So when $x = 0$, from the equation $\frac{0}{3}+\frac{y}{7}=1$, we get $\frac{y}{7}=1$, and $y = 7$. So another point on the line is $(0,7)$.

Step 3: Plot the points and draw the line

• On the coordinate plane, find the point $(3,0)$ (3 units to the right of the origin on the x - axis) and the point $(0,7)$ (7 units up from the origin on the y - axis).
• Use a straight - edge to draw a line passing through these two points.

If we want to write the equation in slope - intercept form ($y=mx + c$, where $m$ is the slope and $c$ is the y - intercept) first:

Step 1: Calculate the slope

The slope $m$ of a line passing through two points $(x_1,y_1)$ and $(x_2,y_2)$ is given by $m=\frac{y_2 - y_1}{x_2 - x_1}$.
We have the points $(3,0)$ and $(0,7)$. Let $(x_1,y_1)=(3,0)$ and $(x_2,y_2)=(0,7)$. Then $m=\frac{7 - 0}{0 - 3}=\frac{7}{-3}=-\frac{7}{3}$.
The y - intercept $c = 7$ (from the given y - intercept). So the equation of the line in slope - intercept form is $y=-\frac{7}{3}x + 7$.

To graph the line using the slope - intercept form:

• Start by plotting the y - intercept $(0,7)$.
• The slope $m =-\frac{7}{3}$ means that for every 3 units we move to the right (increase in $x$ by 3), we move down 7 units (decrease in $y$ by 7). From the point $(0,7)$, if we move 3 units to the right (to $x = 3$) and 7 units down (to $y=7 - 7=0$), we reach the point $(3,0)$, which is the x - intercept. Then draw a line through $(0,7)$ and $(3,0)$.

To answer the question of graphing the line:

1. Plot the points $(3,0)$ (x - intercept) and $(0,7)$ (y - intercept) on the coordinate plane.
2. Draw a straight line passing through these two points.

If we were to describe the graph: The line intersects the x - axis at $(3,0)$ and the y - axis at $(0,7)$. It has a negative slope of $-\frac{7}{3}$, so it is a line that goes from the second quadrant (near the y - axis) to the fourth quadrant (near the x - axis) as $x$ increases.