To solve the system of equations:
-6x + 2y = -32
4x + 3y = -9
We can start by multiplying the first equation by 2 to make the coefficients of y the same:
-12x + 4y = -64
4x + 3y = -9
Now, we can add the two equations together to eliminate y:
-12x + 4y + 4x + 3y = -64 - 9
-8x + 7y = -73
Now we have a single equation with both x and y. We can solve for x by rearranging the equation:
-8x + 7y = -73
-8x = -73 - 7y
x = (73 + 7y)/8
Now we can substitute x back into one of the original equations. Let's use the first equation:
-6(73 + 7y)/8 + 2y = -32
-438/8 - 42y/8 + 2y = -32
-54.75 - 5.25y + 2y = -32
-3.25y = 22.75
y = -7
Now that we have y, we can substitute it back into the equation for x:
x = (73 + 7*(-7))/8
x = (73 - 49)/8
x = 24/8
x = 3
Therefore, the solution to the system of equations is (3, -7).
Question
Solve the system of equations.
−6x+2y=−32
4x+3y=−9 (1 point)
(
,
)
9 answers
Question
The Math Club sold slices of pizza, p, at a profit of $3 each and cans of soda, s, 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.(1 point)
Equation to represent the total number of items sold:
= 200
Equation to represent the total profit:
= 550
The Math Club sold slices of pizza, p, at a profit of $3 each and cans of soda, s, 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.(1 point)
Equation to represent the total number of items sold:
= 200
Equation to represent the total profit:
= 550
Let's represent the number of slices of pizza sold as "x" and the number of cans of soda sold as "y".
To write the pair of linear equations, we can consider the information given:
1. The Math Club sold 200 items in total, so the total number of items sold is 200:
x + y = 200
2. The profit from selling each slice of pizza is $3 and each can of soda is $2, resulting in a total profit of $550:
3x + 2y = 550
Therefore, the pair of linear equations that model this situation are:
1. x + y = 200
2. 3x + 2y = 550
To write the pair of linear equations, we can consider the information given:
1. The Math Club sold 200 items in total, so the total number of items sold is 200:
x + y = 200
2. The profit from selling each slice of pizza is $3 and each can of soda is $2, resulting in a total profit of $550:
3x + 2y = 550
Therefore, the pair of linear equations that model this situation are:
1. x + y = 200
2. 3x + 2y = 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?(1 point)
shirts
shirts
To break even, the revenue from selling the shirts should cover the total cost.
Let's denote:
x = number of shirts sold
Cost = $100 (design and setup) + $10 (cost per shirt) * x (number of shirts sold)
Revenue = $15 * x (selling price per shirt)
Break-even point is when Cost = Revenue:
100 + 10x = 15x
Subtract 10x from both sides:
100 = 5x
Divide by 5:
x = 20
Therefore, the baseball team needs to sell 20 shirts to break even.
Let's denote:
x = number of shirts sold
Cost = $100 (design and setup) + $10 (cost per shirt) * x (number of shirts sold)
Revenue = $15 * x (selling price per shirt)
Break-even point is when Cost = Revenue:
100 + 10x = 15x
Subtract 10x from both sides:
100 = 5x
Divide by 5:
x = 20
Therefore, the 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 the PTA to reach their goal of $3,500?(1 point)
adults;
children
adults;
children
Let's denote:
Let x = number of adults attending the event
Let y = number of children attending the event
Given:
x + y = 500 (Total number of attendees)
10x + 5y = 3500 (Total revenue from admission charges)
We need to solve these two equations simultaneously to find the values of x and y.
From the first equation x + y = 500, we can rearrange it to express y in terms of x:
y = 500 - x
Now substitute this expression for y in the second equation:
10x + 5(500 - x) = 3500
10x + 2500 - 5x = 3500
5x + 2500 = 3500
5x = 1000
x = 200
Now substitute x back into y = 500 - x:
y = 500 - 200
y = 300
Therefore, the PTA needs 200 adults and 300 children to attend for them to reach their goal of $3,500.
Let x = number of adults attending the event
Let y = number of children attending the event
Given:
x + y = 500 (Total number of attendees)
10x + 5y = 3500 (Total revenue from admission charges)
We need to solve these two equations simultaneously to find the values of x and y.
From the first equation x + y = 500, we can rearrange it to express y in terms of x:
y = 500 - x
Now substitute this expression for y in the second equation:
10x + 5(500 - x) = 3500
10x + 2500 - 5x = 3500
5x + 2500 = 3500
5x = 1000
x = 200
Now substitute x back into y = 500 - x:
y = 500 - 200
y = 300
Therefore, the PTA needs 200 adults and 300 children to attend for them to reach their goal of $3,500.
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
y=x+3
−4x+4y=28
To graph the system of linear equations:
1. y = x + 3
2. -4x + 4y = 28
For the first equation, y = x + 3, the slope is 1 (coefficient of x) and the y-intercept is 3. This means the line will pass through the point (0, 3) and has a slope of 1, so it goes up by 1 unit for every 1 unit moved to the right.
For the second equation, we need to rewrite it in slope-intercept form:
-4x + 4y = 28
4y = 4x + 28
y = x + 7
Now, let's compare the two equations. They are both of the form y = mx + b, where m is the slope and b is the y-intercept.
In this case, both equations have the same slope (1), so the lines are parallel. If the lines have the same slope but different y-intercepts, they are parallel and will never intersect. Therefore, there is no solution to this system of linear equations, as the lines never meet.
When you graph the lines, you will see two parallel lines that do not intersect. This tells us that there is no solution that will satisfy both equations simultaneously.
1. y = x + 3
2. -4x + 4y = 28
For the first equation, y = x + 3, the slope is 1 (coefficient of x) and the y-intercept is 3. This means the line will pass through the point (0, 3) and has a slope of 1, so it goes up by 1 unit for every 1 unit moved to the right.
For the second equation, we need to rewrite it in slope-intercept form:
-4x + 4y = 28
4y = 4x + 28
y = x + 7
Now, let's compare the two equations. They are both of the form y = mx + b, where m is the slope and b is the y-intercept.
In this case, both equations have the same slope (1), so the lines are parallel. If the lines have the same slope but different y-intercepts, they are parallel and will never intersect. Therefore, there is no solution to this system of linear equations, as the lines never meet.
When you graph the lines, you will see two parallel lines that do not intersect. This tells us that there is no solution that will satisfy both equations simultaneously.
1 answer