Big Ideas Math Algebra 1 Student Journal 1st Edition Chapter 3 Graphing Linear Functions
Page 74 Exercise 1 Answer
Given: It is given that we sold total $16 tickets. Each adult ticket cost $4 and each child ticket cost $2.
To find We have to find the equation that relates x and y, where x represents the number of adult tickets. Let y represent the number of child tickets.
We will multiply the variable to the corresponding price of tickets and put that equal to the total price of tickets.
Let x represent the number of adult tickets. Let y represent the number of child tickets.
Each adult ticket cost $4 and each child ticket cost $2.
The equation representing the given scenario is
The equation that relates “x” and “y’ is 4x+2y=16.
Read and Learn More Big Ideas Math Algebra 1 Student Journal 1st Edition Solutions
Given: The total cost of tickets and the cost of each adult and child ticket is given to us.
To find We have to fill up the given table.
We will put the value of “x” and get different values of “y”.
Let us put different values of “x” in the equation 4x+2y=16 and get different values of “y”. We get
⇒ x=0
4×0+2y=16
y=8
⇒x=1
4×1+2y=16
2y=12
y=6
⇒x=2
4×2+2y=16
y=4
⇒x=3
4×3+2y=16
2y=4
y=2
⇒x=4
y=16−16
y=0
The different combinations are shown in the table below
The table of combinations is shown below
Given: We have a table that shows the different combinations of tickets we might have sold.
To find We have to plot the points from the table and have to describe the pattern formed by the points.
We will first plot the points after that we will join them and observe the type of plot to tell the pattern formed by these points.
The points from the table are plotted below
The pattern formed by these points is a straight line.
The graph is shown below
The pattern formed by these points is a straight line.
The total amount is $16
The cost of each adult ticket is $4
The cost of each child ticket is $2
The equation representing this case is,
4x+2y=16
where x is the number of adult tickets sold and y is the number of child tickets sold. having the value of x, we can substitute it in the linear equation and only one unknown is present, which we can solve by suitable arithmetic operations.
That is, 2y=16−4x
\(y=\frac{16-4 x}{2}\)Thus we can find the number of child tickets sold.
Given you sold a total of $16 worth of tickets to a fundraiser. The tickets are of two types, adult tickets are $4 each and child tickets are $2 each. If we have the number of adult tickets, x sold we can find the number of child tickets sold, y by substituting the value in the equation,
\(y=\frac{16-4 x}{2}\)
Page 75 Exercise 2 Answer
Given that you sold a total of $48 worth of cheese.
The cost of one swiss cheese per pound is $8
The cost of cheddar cheese per pound is $6
Let the number of pounds of Swiss cheese be x
Let the number of pounds of cheddar cheese be y.
Then we can say that the total cost of sold cheese is the sum of the costs of totals of both the swiss cheese and the cheddar cheese.
Therefore the equation becomes 8x+6y=48.
Given that you sold a total of $48 worth of cheese, where the cost of swiss cheese per pound is $8 and the cost of cheddar cheese per pound is $6. Let x and y represent the number of pounds of Swiss cheese and cheddar cheese.
Then the equation relating x and y is 8x+6y=48.
Given: The equation representing the situation is 8x+6y=48, where x and y
represents the number of pound of swiss cheese and cheddar cheese sold so that the total cost is $48. The coefficients of the variables represent the cost of each of the item.
To find the domain and range of the equation, its graph and also solve for y.
Summary: To solve for isolate the variable in the LHS using suitable arithmetic operations that preserve the equality sign. Choose some values for any of the variables and solve for the unknown. This way find two points plot them on the graph and join them to get the straight line. The domain is the set of all values the variable can take and the range is the set of all output values.
The equation we got is
8x+6y=48
Subtract 8xon both sides
6y=−8x+48
\(y=\frac{-4 x}{3}+8\)
To find two points, choose some values. Put x=0 in the above equation implies y=8.
Put x=3
\(y=\frac{-4}{3} \times 3+8\)y=−4+8
y=4
We have got two points as (0,8)(3,4). Plot the points and join them to obtain the graph.
The domain is the set of all input values of the variables.
The range is the set of all output values.
8x+6y=48
Since x and y are numbers of pounds of swiss and cheddar cheese, the domain is the set of all solutions of the linear equation such that both are whole numbers. The range is only one value, that is {48}.
Given that you sold a total of $48 worth of cheese. Swiss cheese costs $8 per pound.
Cheddar cheese costs $6 per pound.
The equation we get is 8x+6y=48 where x and y represent the number of pounds of swiss cheese and cheddar cheese sold respectively.
Solving for y we get\(y=\frac{-4}{3} x+8\).
The domain of the equation is {(x,y);8x+6y=48,x,y ∈W}, where W is the set of all whole numbers.
The range of the equation is {48}.
The graph of the equation is
In the earlier part of the problem we got the graph of the equation representing this case, that is 8x+6y=48 as
We observe that the point that crosses the x-axis is (6,0) and its x-coordinate is 6, which is the x-intercept. And the point that crosses the y-axis is (0,8) and its y-coordinate is 8, which is the y-intercept.
Given you sold a total of $48 worth of cheese. Swiss cheese costs $8 per pound. Cheddar cheese costs $6 per pound.
The equation we got is 8x+6y=48, where x and y represent the number of pounds of swiss cheese and cheddar cheese sold.
From the graph of the equation, we found that the x-intercept is 6 and the y-intercept is 8. The graph is as follows
The equation we got, in this case, is 8x+6y=48 where x and y represent the number of pounds of swiss cheese and cheddar cheese respectively. Their coefficients represent the cost of each of the corresponding items in dollars.
To find the x-intercept we substitute y=0 in the linear equation, 8x+6y=48, and solve for x.
That is,8x=48
\(x=\frac{48}{8}\)
x=6
To find the y-intercept we substitute x=0 in the equation, 8x+6y=48, and solve for y.
6y=48
\(y=\frac{48}{6}\)
y=8
This is because a point on the x-axis is of the form (x,0), and a point on the y
axis is of the form (0,y).
Given you sold a total of $48worth of cheese, whereas swiss cheese costs $8
per pound and cheddar, cheese costs $6 per pound. This equation representing the case is 8x+6y=48, where x and y represents the number of pounds of swiss cheese and cheddar cheese respectively. The x-intercept is 6 and the y-intercept is 8.
Given that you sold a total of $48 worth of cheese, whereas swiss cheese costs $8
per pound and cheddar, cheese costs $6 per pound.
The equation is 8x+6y=48, where x and y represent the number of pounds of swiss and cheddar cheese respectively.
We found in the earlier part that the x-intercept is 6. This means that if no cheddar cheese was sold then the number of pounds of swiss cheese to be sold so as the total price is $48 is 6.
Similarly, the y-intercept is 8. This says that if no amount of swiss cheese was sold then 8 pounds of cheddar cheese has to be sold so that the total cost is $48.
Given you sold a total of $48 worth of cheese, whereas swiss cheese costs $8
per pound and cheddar, cheese costs $6 per pound. The equation representing the case is 8x+6y=48, where x and y represent the number of pounds of swiss cheese and cheddar cheese sold respectively. The x and y intercepts are 6 and 8 respectively. This means if no amount of cheddar cheese was sold then 6 pounds of swiss cheese has to be sold and if no amount of swiss cheese was sold then 8 pounds of cheddar cheese has to be sold so that the total cost is $48.
Page 75 Exercise 3 Answer
The equation Ax+By=C represents a straight line, provided A, B are not both zero.
If B=0 and A≠0 then the equation is of the form x=\(\frac{C}{A}\)
; constant which represents a vertical line that is a line parallel to the y-axis.
If A=0 and B≠0 then the equation is of the form y=\(\frac{C}{B}\)
; constant which represents a horizontal line, that is a line parallel to the x-axis.
If both A≠0 B≠0, then the equation can be written as
By=−Ax+C
y=−\(\frac{A}{B}\)x+\(\frac{C}{B}\)
This is the slope-intercept form of the linear equation. The graph is a straight line with slope as the coefficient of x and the y-intercept is the constant.
The equation, Ax+By=C is the standard form a linear equation. The graph of the equation is a straight line. It can be a horizontal line or a vertical line or a line with a particular slope which intersects the axes at some points.
Page 77 Exercise 2 Answer
To draw the line x=2, we choose two points of the form (2,y), say, (2,−2),(2,5). Plot these points on the graph and join them.
The graph of the linear equation, x=2, is
Page 78 Exercise 5 Answer
Given: The linear equation is 5x−2y=−30.
To find The x and y intercepts of the graph of the linear equation.
Summary: For a point on the x-axis, the y coordinate is 0 and for a point on the y-axis, the x coordinate is 0. To find the x-intercept put y=0 and solve for x.
Similarly to find the y-intercept put x=0 in the linear equation and solve for y.
Put y=0
In 5x−2y=−30
5x=−30
x=−30
\(x=\frac{-30}{5}\)x=−6
The x-intercept.
Put x=0
In 5x−2y=−30
−2y=−30
\(y=\frac{-30}{-2}\)y=15
The y-intercept is 15.
The x and y intercepts of the linear equation, 5x−2y=−30 is −6, and 15 respectively.
Page 78 Exercise 6 Answer
Given: The linear equation is −8x+12y=24.
To find the x and y intercepts and thus draw the graph.
Summary: To find the x-intercept put y=0 and solve for x. Similarly to find the y-intercept put x=0 in the linear equation and solve for y.
Plot the points on the axes and join them to obtain the graph.
To find the intercepts
Put y=0
In −8x+12y=24.
−8x=24
\(x=\frac{24}{-8}\)x=−3
The point at which the line crosses the x-axis is (−3,0).
Put x=0
In −8x+12y=24.
12y=24
\(y=\frac{24}{12}\)y=2
The point at which the line intersects the y-axis is (0,2).
Having got both the intercepts, plot the points, (−3,0,)(0,2) on the graph and join them.
The x and y intercepts of the linear equation, −8x+12y=24 is −3 and 2 respectively. And its graph is as follows
Page 78 Exercise 7 Answer
Given: The linear equation is 2x+y=4.
To find the x and y intercepts of the line and thus draw the graph of the equation.
Summary: First find the intercepts. Put y=0 in the linear equation and solve for x to obtain the x-intercept.
Similarly, put x=0 in the equation and solve for y to obtain the y-intercept. Plot the point on their respective axes. Join them to get the graph.
Let us find the intercepts.
Put y=0
In 2x+y=4.
2x=4
\(x=\frac{4}{2}\)x=2
The point that intersects the x-axis is (2,0).
Put x=0 in 2x+y=4.
2×0+y=4
y=4
The point that intersects the y-axis is (0,4).
Plot the points, (2,0), and (0,4) on the axis and join them to obtain the graph.
The x and y intercepts of the linear equation, 2x+y=4 are 2 and 4 respectively. And the graph of the equation is as follows.
Page 78 Exercise 8 Answer
Given: The equation, 25x+10y=9000, where x is the number of sweatshirts sold and y is the number of baseball caps sold. And the coefficients are the cost of each item corresponding to the variable in dollars.
To find the intercepts of the linear equation and interpret them.
Summary: Putting y=0 in the equation and solving for x gives us the x-intercept and Putting x=0 in the equation and solving for y gives us the y-intercept. We can interpret the result in terms of the items and the cost that the variables represent.
Put y=0
In 25x+10y =9000.
25x=9000
\(x=\frac{9000}{25}\)x=360
The point that crosses the x-axis is(360,0).
Put x=0
In 25x+10y=9000.
10y=9000
\(y=\frac{9000}{10}\)=900
The point that intersects the y-axis is(0,900).
The variables represent the following
x=The number of sweatshirts sold.
25=The cost of one sweatshirt in dollars.
y=The number of baseball caps sold.
10=The cost of one baseball cap.
9000=Total amount to be collected to attend the band competition.
The point (360,0) tells us that the number of sweatshirts sold if no baseball caps were sold is 360.
The point (0,900) tells us that the number of baseball caps to be sold if no sweatshirts were sold is 900.
The equation is 25x+10y=9000, where x and y represent the number of sweatshirts and baseballs to be sold so as to get the amount of 9000 dollars to attend a band competition. Also, the coefficients 25 and 10 represent the amount of each sweatshirt and baseball cap in dollars.
The x and y intercepts are 360 and 900 respectively. This says that the number of sweatshirts sold when no baseball caps were sold is 360 and the number of baseball caps to be sold when no sweatshirts were sold is 900, so as to achieve the target amount.
Given: the equation 25x+10y=9000, are the number of sweatshirts and baseballs sold and their coefficients represent the cost of respective item in dollars.
To find the number of baseball caps sold if the number of sweatshirts sold is 258.
Summary: Substitute x=258 in the given linear equation and solve for y.
Put x=258
In 25x+10y=9000.
25×258+10y=9000
6450+10y=9000
Subtract 6450 on both sides of the equation.
10y=9000−6450
10y=2550
\(y=\frac{2550}{10}\)y=225
which is the number of baseball caps sold.
Given: The equation 25x+10y=9000 models the given situation, where x and y represent the number of sweatshirts and baseball caps sold so that the total amount collected is $9000.
To find the graph of the solution and two more possible solutions in this context.
Summary: Choose some values for the variables and solve for the other. Thus obtain two points. Plot and join them to get the graph.
To find the solutions in this context find out solutions of the equation so that they are whole numbers since the variables represent a number.
In part (a) of the problem we found that the x and y intercepts of the equation, 25x+10y=9000 are 360,900 respectively.
So two points on the line are (360,0), and (0,900). Plot the points and join them.
Since the variables represent a number of items sold, the domain of the equation, 25x+10y=9000 is {(x,y);25x+10y=9000,x,y ∈W}, where W is the set of all whole numbers.
The equation can be written as
10y=9000−25x
Let x=2⇒
\(y=\frac{9000-25 \times 2}{10}\) \(=\frac{8950}{10}\)=895
Let x=4⇒
\(y=\frac{9000-25 \times 2}{10}\) \(=\frac{8900}{10}\)=890
If the number of sweatshirts sold is 2 then the number of baseball caps sold is 895. If the number of sweatshirts sold is 4 then the number of baseball caps sold is 890.
The equation 25x+10y=9000 represents the situation, where x and y represent the number of sweatshirts and baseball caps sold so that the total amount collected is $9000. Two possible solutions are (2,895), and (4,890). The graph of the equation is.
