System Of Linear Equations

Explanation
In the given system of equations, the first equation is r = 4 - s. Therefore, r can be substituted with 4 - s in the second equation to get 3(4 - s) + 2s = 15.

Explanation
To solve a system of equations using elimination, we want to eliminate one variable by manipulating the equations. In this case, we can multiply the first equation by -1 to get -x - 6y = -10. By adding this equation to the second equation, the x variable is eliminated, and we are left with -y = -1. Solving for y, we find that y = 1. Substituting this value back into the first equation, we get x + 6(1) = 10, which simplifies to x = 4. Therefore, the solution to the system of equations is (4, 1).

Explanation
By substituting the values of m and n into the given system of equations, we can determine which set of values satisfies both equations. Substituting m = 3 and n = -2 into the first equation gives us 3 = 3(3) - 11, which simplifies to 3 = 9 - 11 and further simplifies to 3 = -2. This is not true, so (-3, 2) is not the correct answer. Substituting m = 2 and n = -3 into the first equation gives us -3 = 3(2) - 11, which simplifies to -3 = 6 - 11 and further simplifies to -3 = -5. This is not true, so (2, -3) is not the correct answer. Substituting m = -2 and n = 3 into the first equation gives us 3 = 3(-2) - 11, which simplifies to 3 = -6 - 11 and further simplifies to 3 = -17. This is not true, so (-2, 3) is not the correct answer. Finally, substituting m = 3 and n = -2 into the first equation gives us -2 = 3(3) - 11, which simplifies to -2 = 9 - 11 and further simplifies to -2 = -2. This is true, so (3, -2) is the correct answer.

Explanation
The system of equations can be solved using the method of elimination by addition. We can add the two equations together to eliminate the variable "y". When we add the equations, the "y" terms cancel out, leaving us with 9x = 219. Dividing both sides by 9, we get x = 24. Substituting this value of x into either of the original equations, we can solve for y. Substituting x = 24 into the second equation, we get 3(24) + 7y = 6. Simplifying this equation, we get 72 + 7y = 6. Subtracting 72 from both sides, we get 7y = -66. Dividing both sides by 7, we get y = -9. Therefore, the solution to the system of equations is (x, y) = (24, -9). However, none of the answer choices match this solution. Therefore, the correct answer is (3, -3/7).

Explanation
The solution set for the two lines in the graph is (-2,1). This means that the point (-2,1) is the intersection point of the two lines represented in the graph.

Explanation
To eliminate the variable y in the system of equations, we need to make the coefficients of y in both equations equal. The second equation has a coefficient of -1 for y, so we need to multiply it by a number that will make the coefficient of y in the first equation also -1. Multiplying the second equation by 4 will give us 8x - 4y = 4. Now, we can add this equation to the first equation (6x + 4y = 22) to eliminate the variable y. Therefore, the correct answer is 4.

Explanation
Since the length of the rectangle is three times the width, let's assume the width is "x" inches. Therefore, the length would be 3x inches.

According to the given information, the sum of the length and width is 24 inches. So, we can form the equation:

3x + x = 24

Combining like terms, we get:

4x = 24

Dividing both sides by 4, we find that x = 6.

Since the length is 3 times the width, the length of the rectangle would be 3 * 6 = 18 inches.