2. Use an augmented matrix to solve the system.
x + y = 5
3x – y = –1 (1 point)
(1, 4)
(1, 5)
(3, –1)***
(5, –1)
3. When converting a system of linear equations into an augmented matrix, what equation form is needed? (1 point)
slope-intercept form
negative form
graph form***
standard form
4. What does the vertical line in an augmented matrix represent? (1 point)
x-value
y-value***
equal sign
line on a graph
5. You have $1.00 in dimes and nickels. You have twice as many dimes as nickels. What would the associated system of equations look like in augmented matrix form? (1 point)
matrix 1***
matrix 2
matrix 3
matrix 4
Please help!
first four wrong, appears you are just guessing.
on the last, isn't this the system?
10d+5n=100
d-2n=0
is that a matrix 1 system?
No I just realized that. I did the system wrong. I'm dumb :/ That would be a MATRIX 3
2. To solve the system using an augmented matrix, we set up the augmented matrix by writing the coefficients of the variables and the constants on the right side of the equation. Then we perform row operations to row-reduce the augmented matrix.
The given system of equations is:
x + y = 5 (equation 1)
3x - y = -1 (equation 2)
To set up the augmented matrix, we write the coefficients of the variables and the constants on the right side:
| 1 1 | 5 |
| 3 -1 | -1 |
Now we perform row operations to obtain row-echelon form:
Row 2 - 3*(Row 1) (multiply row 1 by -3 and add it to row 2)
| 1 1 | 5 |
| 0 -4 | -16 |
We can see that row 2 represents the equation -4y = -16, which simplifies to y = 4. By substituting this value of y into equation 1, we can solve for x:
x + 4 = 5
x = 1
Therefore, the solution to the system is (x, y) = (1, 4).
The correct answer is (1, 4).
3. When converting a system of linear equations into an augmented matrix, the equation form needed is the standard form. This form is written as follows:
Ax + By = C
The correct answer is standard form.
4. In an augmented matrix, the vertical line represents the equal sign. It separates the coefficients of the variables on the left side from the constants on the right side.
The correct answer is the equal sign.
5. To represent the given problem in augmented matrix form, we first need to express the information as a system of linear equations.
Let's assume:
x = number of dimes
y = number of nickels
According to the problem, we have $1.00 in dimes and nickels. Therefore, we can write the equation for the total amount as:
0.10x + 0.05y = 1.00 (equation 1)
We are also given that we have twice as many dimes as nickels. This can be expressed as:
x = 2y (equation 2)
To convert this system of linear equations into augmented matrix form, we write the coefficients of the variables and the constant on the right side:
| 0.10 0.05 | 1.00 |
| 1 -2 | 0 |
This represents the given system of equations in augmented matrix form.
The correct answer is matrix 1.
2. To solve the system using an augmented matrix, we can set up the coefficients of the variables on the left side of the matrix, and the constants on the right side. The augmented matrix would look like this:
[1 1 | 5]
[3 -1 | -1]
Now, we can perform row operations on the matrix to simplify it and solve for the variables. Let's use the Gaussian elimination method:
Step 1: Multiply the first row by 3 and subtract it from the second row to eliminate the x-term in the second equation:
[1 1 | 5]
[0 -4 | -16]
Step 2: Divide the second row by -4 to make the coefficient of y equal to 1:
[1 1 | 5]
[0 1 | 4]
Step 3: Subtract the second row from the first row to eliminate the y-term in the first equation:
[1 0 | 1]
[0 1 | 4]
The solution is x = 1 and y = 4, so the correct answer is (1, 4).
3. When converting a system of linear equations into an augmented matrix, the equation form needed is the standard form. This form is written as:
Ax + By = C
Where A, B, and C are coefficients or constants, and x and y represent the variables.
4. In an augmented matrix, the vertical line represents the equal sign. It separates the coefficients of the variables on the left side of the line from the constants on the right side.
5. In this case, you have $1.00 in dimes and nickels, with twice as many dimes as nickels. Let's assign the variable x to represent the number of nickels and y to represent the number of dimes. The associated system of equations is:
x + y = 10 (since there are 10 coins in total)
0.05x + 0.10y = 1.00 (since a nickel is worth $0.05 and a dime is worth $0.10)
To convert this system of equations into augmented matrix form, we arrange the coefficients of the variables and the constants in a matrix:
[1 1 | 10]
[0.05 0.10 | 1.00]
So, the correct answer is "matrix 1."
I hope this explanation helps you understand the process of solving systems of equations using augmented matrices!