Review the graphs of a system of these two linear equations in two variables: y = 4x - 10 and y = - 2x + 8 FInd the solution of system.
The solution of the system of equation is (__).
To find the solution of the system of equations, we need to find the values of x and y that satisfy both equations simultaneously.
First, let's graph each equation to see where they intersect.
For the equation y = 4x - 10, we can plot a few points to create a line:
When x = 0, y = 4(0) - 10 = -10. So we have the point (0, -10).
When x = 2, y = 4(2) - 10 = -2. So we have the point (2, -2).
When x = 4, y = 4(4) - 10 = 6. So we have the point (4, 6).
Plotting these points and drawing a line through them, we get:
Graph 1: Line representing y = 4x - 10
Next, for the equation y = -2x + 8, we can again plot points to create a line:
When x = 0, y = -2(0) + 8 = 8. So we have the point (0, 8).
When x = 2, y = -2(2) + 8 = 4. So we have the point (2, 4).
When x = 4, y = -2(4) + 8 = 0. So we have the point (4, 0).
Plotting these points and drawing a line through them, we get:
Graph 2: Line representing y = -2x + 8
Now, we can see that the two lines intersect at the point (3, 2). This means that the values of x and y that satisfy both equations simultaneously are x = 3 and y = 2.
Therefore, the solution of the system of equations is (3, 2).
Estimate the solution to the system of these linear equations based on the graph.
y = 2x + 3
y = 1/2x + 6
(2, 7)
(7, 2)
(2, -7)
(-2, 7)
Based on the graph of the system of linear equations, it appears that the lines representing the equations y = 2x + 3 and y = 1/2x + 6 intersect at the point (2, 7).
Therefore, the estimate for the solution to the system of equations is (2, 7).
Estimate the solution to the system of these linear equations based on the graph.
y = -x + 6
y = 3x - 2
(__)
Based on the graph of the system of linear equations, it appears that the lines representing the equations y = -x + 6 and y = 3x - 2 intersect at the point (2, 4).
Therefore, the estimate for the solution to the system of equations is (2, 4).
Determine the number of solutions to this system of linear equations:
2x + y = 6
4x + 2y = 8
1
infinite
2
0
To determine the number of solutions to the system of linear equations, we can analyze the system using various methods such as graphing, substitution, or elimination.
Let's use the elimination method:
We can multiply the first equation by 2 to make the coefficients of y in both equations the same:
2(2x + y) = 2(6)
4x + 2y = 12
Now we compare this equation to the second equation:
4x + 2y = 8
Since both equations have the same coefficients for x and y terms, we can subtract the first equation from the second equation:
(4x + 2y) - (4x + 2y) = 8 - 12
0 = -4
This equation simplifies to 0 = -4, which is not a true statement. This means that there are no values of x and y that satisfy both equations simultaneously.
Therefore, the system of linear equations has 0 solutions, or is inconsistent.
Which of the graphs with no solution correctly shows the system of equations
y = 2x + 4 and - 2x + y = 8?
The correct graph that shows the system of equations y = 2x + 4 and -2x + y = 8 with no solution is the one where the two lines are parallel and do not intersect.
In this case, the lines have the same slope of 2, but different y-intercepts. This means that they will never intersect, and there is no solution that satisfies both equations simultaneously.
Therefore, the correct graph with no solution would show two parallel lines.
Which of these graphs shows that the linear system 3x + 2y = 12
and 6x + 4y = 24 has an infinite number of solutions?
The graph that shows the linear system 3x + 2y = 12 and 6x + 4y = 24 has an infinite number of solutions is the one where the two lines representing the equations are the same line.
In this case, the two equations are actually multiples of each other. If we simplify the second equation, we get:
6x + 4y = 24
Dividing both sides by 2, we have:
3x + 2y = 12
By comparing the two equations, we can see that they are equivalent. This means that the equations represent the same line and all points on this line are solutions to both equations.
Therefore, the correct graph that shows infinite solutions would be one where the two lines overlap and coincide with each other.
Solve the system of equations.
4x - 8y = -36
-6x + y = 32
To solve the system of equations:
1) Multiply the second equation by 8 to make the coefficients of y in both equations the same:
8(-6x + y) = 8(32)
-48x + 8y = 256
So now we have the system of equations:
4x - 8y = -36
-48x + 8y = 256
2) Now we can add the two equations together to eliminate the y terms:
(4x - 8y) + (-48x + 8y) = -36 + 256
-44x = 220
Divide both sides by -44 to solve for x:
x = -220/-44
x = 5
3) Now substitute the value of x back into either of the original equations and solve for y. Let's use the first equation:
4x - 8y = -36
4(5) - 8y = -36
20 - 8y = -36
Subtract 20 from both sides:
-8y = -56
Divide both sides by -8 to solve for y:
y = -56/-8
y = 7
So the solution to the system of equations is x = 5 and y = 7
Solve the system of equations.
6x - 3y = -6
9x- 3y = -18
To solve the system of equations:
1) Notice that both equations have the same coefficient for y. We can subtract the second equation from the first equation to eliminate the y term:
(6x - 3y) - (9x - 3y) = -6 - (-18)
6x - 3y - 9x + 3y = -6 + 18
-3x = 12
2) Divide both sides of the equation by -3 to solve for x:
x = 12 / -3
x = -4
3) Now substitute the value of x back into either of the original equations and solve for y. Let's use the first equation:
6x - 3y = -6
6(-4) - 3y = -6
-24 - 3y = -6
Subtract -24 from both sides:
-3y = 18
Divide both sides by -3 to solve for y:
y = 18 / -3
y = -6
So the solution to the system of equations is x = -4 and y = -6.
Solve the system of equations.
6x - 3y = -6
9x- 3y = -18
(__, __)
To solve the system of equations:
We can see that both equations have the same coefficient for y. By subtracting the second equation from the first equation, we can eliminate the y term:
(6x - 3y) - (9x - 3y) = -6 - (-18)
6x - 3y - 9x + 3y = -6 + 18
-3x = 12
We divide both sides of the equation by -3 to solve for x:
x = 12 / -3
x = -4
Now we can substitute the value of x back into either of the original equations to solve for y. Let's use the first equation:
6x - 3y = -6
6(-4) - 3y = -6
-24 - 3y = -6
We can add 24 to both sides:
-3y = -6 + 24
-3y = 18
Dividing both sides by -3:
y = 18 / -3
y = -6
Therefore, the solution to the system of equations is (-4, -6).
Solve the system of equations.
5x-4y=-44
- x + 3y = 22
(__, __)
To solve the system of equations:
We can solve this system of equations using either the substitution or the elimination method. Let's use the elimination method:
Multiply the second equation by 5 to make the coefficients of x the same in both equations:
5(-x + 3y) = 5(22)
-5x + 15y = 110
Now, we can add the two equations together to eliminate the x term:
(5x - 4y) + (-5x + 15y) = -44 + 110
11y = 66
Divide both sides of the equation by 11 to solve for y:
y = 66 / 11
y = 6
Now substitute the value of y back into either of the original equations to solve for x. Let's use the first equation:
5x - 4y = -44
5x - 4(6) = -44
5x - 24 = -44
Add 24 to both sides:
5x = -20
Divide both sides by 5 to solve for x:
x = -20 / 5
x = -4
Therefore, the solution to the system of equations is (-4, 6).
Solve the system of equations.
-6x + 2y = -32
4x + 3y = -9
(__, __)
To solve the system of equations:
We can solve the system of equations using either the substitution or the elimination method. Let's use the elimination method.
First, we can multiply the first equation by 4 and the second equation by 6 to make the coefficients of x the same in both equations:
4(-6x + 2y) = 4(-32)
6(4x + 3y) = 6(-9)
Simplifying these equations, we get:
-24x + 8y = -128
24x + 18y = -54
Now, we can add the two equations together to eliminate the x term:
(-24x + 8y) + (24x + 18y) = -128 + (-54)
26y = -182
Divide both sides of the equation by 26 to solve for y:
y = -182 / 26
y = -7
Now substitute the value of y back into either of the original equations to solve for x. Let's use the first equation:
-6x + 2y = -32
-6x + 2(-7) = -32
-6x - 14 = -32
Add 14 to both sides:
-6x = -18
Divide both sides by -6 to solve for x:
x = -18 / -6
x = 3
Therefore, the solution to the system of equations is (3, -7).
The Math Club sold slices of pizza, p, at a profit of $3 each and cans of soda, 8, at $2 each to raise money for a trip. They sold 200 items and made a profit of $550. Write the pair of linear equations that model this situation.
Equation to represent the total number of items sold: __ = 200
Equation to represent the total profit: __ = 550
Equation to represent the total number of items sold: p + 8 = 200
In this equation, p represents the number of slices of pizza sold and 8 represents the number of cans of soda sold. The total number of items sold is equal to 200.
Equation to represent the total profit: 3p + 2(8) = 550
In this equation, 3p represents the profit from selling pizza slices (3 dollars profit per slice) and 2(8) represents the profit from selling cans of soda (2 dollars profit per can). The total profit is equal to 550 dollars.
please answer the question
Equation to represent the total number of items sold: p + 8 = 200
Equation to represent the total profit: 3p + 2(8) = 550
The Lakewood baseball team is selling T-shirts for a fundraiser. The shirts cost $100 for the printing design and setup, plus $10 per shirt. The team is going to sell the shirts for $15 each. How many shirts do they need to sell to break even?
__ shirts
To determine how many shirts the Lakewood baseball team needs to sell to break even, we can set up a linear equation.
Let x represent the number of shirts sold.
The total cost to produce the shirts is $100 for the printing design and setup, plus $10 per shirt. Therefore, the total cost is 100 + 10x.
The revenue generated by selling the shirts is $15 per shirt, so the total revenue is 15x.
To break even, the total cost and total revenue should be equal. So we can set up the equation:
100 + 10x = 15x
Now we can solve for x.
Subtract 10x from both sides of the equation:
100 = 5x
Divide both sides by 5:
20 = x
Therefore, the Lakewood baseball team needs to sell 20 shirts to break even.
The Kesling Middle School PTA is planning a carnival to raise money for the school's art department. They estimate that the event will be very popular and that they will have 500 people attend. They plan to charge adults $10 and children $5 for admission. The PTA wants to earn $3,500 from admission charges. How many adults and how many children need to attend for PTA to reach their goal of $3,500?
__ adults; __ children
Let's use the variables a for the number of adults and c for the number of children attending the carnival.
We are given the following information:
- The total number of people attending the carnival is 500: a + c = 500.
- The PTA wants to earn $3,500 from the admission charges: 10a + 5c = 3500.
To solve this system of equations, we can use substitution or elimination. Let's use the elimination method.
Multiply the first equation by 5 to make the coefficients of c the same in both equations:
5(a + c) = 5(500)
5a + 5c = 2500.
Now we can subtract the second equation from this new equation:
(5a + 5c) - (10a + 5c) = 2500 - 3500
-5a = -1000.
Divide both sides of the equation by -5:
a = -1000 / -5
a = 200.
Now substitute the value of a back into the first equation to solve for c:
200 + c = 500
c = 500 - 200
c = 300.
Therefore, the PTA needs 200 adults and 300 children to attend the carnival to reach their goal of $3,500.
answer the question
The Kesling Middle School PTA needs 200 adults and 300 children to attend the carnival in order to reach their goal of earning $3,500 from admission charges.
You are designing a simulation in which two characters moving along different paths are going to collide when the characters are at the same position at the same time. The position of the first character is given by y = 2t, and the second is y = t + 2, y is in centimeters and t is the time in seconds. How many seconds after the ismulation is started will the characters collide?
4 seconds
1 second
3 seconds
2 seconds
To find the time at which the two characters will collide, we need to set their positions equal to each other and solve for t.
Setting y = 2t and y = t + 2 equal to each other:
2t = t + 2
Subtracting t from both sides:
t = 2
Therefore, after 2 seconds, the characters will collide.
Solve the following system of linear equations by graphing. Graph on your own piece of paper. In your submitted answer, describe what the graph looks like and what this tells you about the solution to the system of linear equations.
y = x + 3
-4x + 4y = 28
To solve the system of linear equations by graphing, we can start by rearranging the second equation into slope-intercept form (y = mx + b).
-4x + 4y = 28
4y = 4x + 28
y = x + 7
Now we have both equations in slope-intercept form:
y = x + 3
y = x + 7
We can then plot points on a graph for both equations and connect them to form lines.
For the equation y = x + 3