1. Solve the system of two linear equations.−3x−4y=−23 3x−5y=5 (1 point)
A. (−5, 2) B. (−5,−2) C. (5, 2) D. (5,−2)
2. How many solutions does the system of equations have?8x−6y=48 8x+6y=48 (1 point)
A.None B. Two C. one D. Infinite
3. Solve the system of two equations using elimination with multiplication. 4x−6y=−48 −2x+4y=30 (1 point)
A.(3, 9) B.(−33,−9) C.(−21,−6) D.(−3, 6)
4. Solve the system of equations.3x−4y=40 5x−5y=55 (1 point)
A,(4,−7) B.(18, 7) C.(8,−3) D.(14, 3)
5. How many solutions does the system have?12x−15y=18 4x−5y=6 (1 point)
A. Two solutions B. one solution C. No solutions D. infinity Solutions
1. To solve the system of equations −3x−4y=−23 and 3x−5y=5, we can use the method of elimination. Multiply the first equation by 3 and the second equation by -3 to eliminate the x term:
-3(-3x-4y) = -3(-23)
9x + 12y = 69
-3(3x-5y) = -3(5)
-9x + 15y = -15
Adding these two equations together, we eliminate the x term:
(9x + 12y) + (-9x + 15y) = 69 + (-15)
27y = 54
y = 2
Substitute this value of y back into either of the original equations, let's use the second equation:
3x - 5(2) = 5
3x - 10 = 5
3x = 15
x = 5
The solution to the system of equations is (5, 2).
Answer: C. (5, 2)
2. To determine the number of solutions of the system of equations 8x-6y=48 and 8x+6y=48, notice that both equations are equivalent when you multiply the second equation by -1:
8x-6y = 48
8x+(-6y) = 48
Since the equations are equivalent, they represent the same line. This means that the system of equations has an infinite number of solutions.
Answer: D. Infinite
3. To solve the system of equations 4x-6y=-48 and -2x+4y=30 using elimination with multiplication, first multiply the second equation by 2:
2(-2x+4y) = 2(30)
-4x + 8y = 60
Now add this equation to the first equation to eliminate the x term:
(4x - 6y) + (-4x + 8y) = -48 + 60
2y = 12
y = 6
Substitute this value of y back into either of the original equations, let's use the first equation:
4x - 6(6) = -48
4x - 36 = -48
4x = -12
x = -3
The solution to the system of equations is (-3, 6).
Answer: D. (-3, 6)
4. To solve the system of equations 3x-4y=40 and 5x-5y=55, we can use the method of elimination. Multiply the first equation by 5 and the second equation by 3 to eliminate the x term:
5(3x-4y) = 5(40)
15x - 20y = 200
3(5x-5y) = 3(55)
15x - 15y = 165
Subtracting these two equations, we eliminate the x term:
(15x - 20y) - (15x - 15y) = 200 - 165
-5y = 35
y = -7
Substitute this value of y back into either of the original equations, let's use the first equation:
3x - 4(-7) = 40
3x + 28 = 40
3x = 12
x = 4
The solution to the system of equations is (4, -7).
Answer: A. (4, -7)
5. To determine the number of solutions of the system of equations 12x-15y=18 and 4x-5y=6, notice that both equations are equivalent when you divide the first equation by 3 and divide the second equation by 2:
(1/3)(12x-15y) = (1/3)(18)
4x - 5y = 6
(1/2)(4x-5y) = (1/2)(6)
2x - (5/2)y = 3
The two resulting equations have the same slope but different y-intercepts. This means that the lines represented by the equations are parallel and will never intersect. Therefore, the system of equations has no solutions.
Answer: C. No solutions
1. To solve the system of equations −3x − 4y = −23 and 3x − 5y = 5 using the elimination method, first multiply the second equation by 3 to make the coefficients of x in both equations opposite each other:
−3x − 4y = −23 (Equation 1)
3x − 5y = 5 (Equation 2)
Multiply Equation 2 by 3:
9x − 15y = 15 (Equation 3)
Next, add Equation 1 and Equation 3 together to eliminate the x terms:
(-3x - 4y) + (9x - 15y) = (-23) + 15
6x - 19y = -8 (Equation 4)
Now, we have a new equation:
6x - 19y = -8 (Equation 4)
Solve Equation 4 for x:
6x = 19y - 8
x = (19y - 8)/6
Now, substitute the expression for x into Equation 1:
-3((19y - 8)/6) - 4y = -23
Multiply both sides by 6 to get rid of the denominator:
-3(19y - 8) - 24y = -138
-57y + 24 - 24y = -138
-81y = -162
y = (-162)/(-81)
y = 2
Substitute y = 2 into the expression for x:
x = (19(2) - 8)/6
x = (38 - 8)/6
x = 30/6
x = 5
Therefore, the solution to the system of equations is (5, 2).
So, the answer is C. (5, 2)
2. To determine the number of solutions the system of equations 8x - 6y = 48 and 8x + 6y = 48 has, we can see that the coefficients of y are opposite each other and the constant terms are the same. This means that the lines represented by these equations are parallel and will never intersect. Therefore, the system has no solution.
So, the answer is A. None
3. To solve the system of equations 4x - 6y = -48 and -2x + 4y = 30 using the elimination method with multiplication, multiply the first equation by 2 to make the coefficients of x in both equations opposite each other:
2(4x - 6y) = 2(-48)
8x - 12y = -96 (Equation 1)
Now, we have the system of equations:
8x - 12y = -96 (Equation 1)
-2x + 4y = 30 (Equation 2)
Next, add Equation 1 and Equation 2 together to eliminate the x terms:
(8x - 12y) + (-2x + 4y) = -96 + 30
6x - 8y = -66 (Equation 3)
Now, we have a new equation:
6x - 8y = -66 (Equation 3)
Solve Equation 3 for x:
6x = 8y - 66
x = (8y - 66)/6
Now, substitute the expression for x into Equation 1:
8((8y - 66)/6) - 12y = -96
Multiply both sides by 6 to get rid of the denominator:
((64y - 528)/6) - 12y = -96
Multiply both sides by 6 to clear the fraction:
64y - 528 - 72y = -576
-8y - 528 = -576
-8y = -576 + 528
-8y = -48
y = (-48)/(-8)
y = 6
Substitute y = 6 into the expression for x:
x = (8(6) - 66)/6
x = (48 - 66)/6
x = (-18)/6
x = -3
Therefore, the solution to the system of equations is (-3, 6).
So, the answer is D. (-3, 6)
4. To solve the system of equations 3x - 4y = 40 and 5x - 5y = 55, we can use the method of substitution.
First, solve the second equation for x:
5x - 5y = 55
5x = 55 + 5y
x = (55 + 5y)/5
x = 11 + y
Now, substitute x = 11 + y into the first equation:
3(11 + y) - 4y = 40
33 + 3y - 4y = 40
-y = 7
y = -7
Substitute y = -7 into x = 11 + y:
x = 11 + (-7)
x = 4
Therefore, the solution to the system of equations is (4, -7).
So, the answer is A. (4, -7)
5. To determine the number of solutions for the system of equations 12x - 15y = 18 and 4x - 5y = 6, we can compare the slopes of the lines represented by these equations.
The slopes of the two lines are equal since the coefficients of x and y in both equations are directly proportional. Therefore, the lines are parallel and will never intersect.
Since the lines are parallel, the system has no solutions.
So, the answer is C. No solutions.
1. To solve the system of equations:
-3x - 4y = -23 ...(equation 1)
3x - 5y = 5 ...(equation 2)
One way to solve this system is by using the method of substitution.
First, let's isolate one variable in terms of the other from one of the equations. From equation 1, we can isolate x:
-3x = -23 + 4y
Divide both sides by -3:
x = (23/3) - (4/3)y
Now substitute this value of x into equation 2:
3((23/3) - (4/3)y) - 5y = 5
Simplify:
23 - 4y - 5y = 5
Combine like terms:
-9y = -18
Divide both sides by -9:
y = 2
Now substitute this value of y back into the expression for x:
x = (23/3) - (4/3)(2)
x = 23/3 - 8/3
x = 15/3
x = 5
The solution to the system of equations is (5, 2).
Therefore, the correct answer is C. (5, 2).
2. To determine how many solutions the system of equations has:
8x - 6y = 48 ...(equation 1)
8x + 6y = 48 ...(equation 2)
One way to determine the number of solutions is by comparing the slopes of the two lines formed by the equations.
Both equations have the same slope of 8/6, which simplifies to 4/3.
When two lines have the same slope and different y-intercepts, they intersect at a single point, meaning they have one unique solution.
Therefore, the correct answer is C. one.
3. To solve the system of equations using elimination with multiplication:
4x - 6y = -48 ...(equation 1)
-2x + 4y = 30 ...(equation 2)
One way to solve this system is by using the method of elimination with multiplication.
We can eliminate the x variable by multiplying equation 1 by -2 and equation 2 by 4.
-2(4x - 6y) = -2(-48)
4(-2x + 4y) = 4(30)
Expanding and simplifying, we have:
-8x + 12y = 96 ...(equation 3)
-8x + 16y = 120 ...(equation 4)
Now we can subtract equation 3 from equation 4 to eliminate the x variable:
-8x + 16y - (-8x + 12y) = 120 - 96
-8x + 16y + 8x - 12y = 24
4y = 24
Divide both sides by 4:
y = 6
Now substitute this value of y into equation 1 to solve for x:
4x - 6(6) = -48
4x - 36 = -48
4x = -48 + 36
4x = -12
Divide both sides by 4:
x = -12/4
x = -3
The solution to the system of equations is (-3, 6).
Therefore, the correct answer is D. (-3, 6).
4. To solve the system of equations:
3x - 4y = 40 ...(equation 1)
5x - 5y = 55 ...(equation 2)
One way to solve this system is by using the method of substitution.
First, let's isolate one variable in terms of the other from one of the equations. From equation 1, we can isolate x:
3x = 40 + 4y
Divide both sides by 3:
x = (40/3) + (4/3)y
Now substitute this value of x into equation 2:
5((40/3) + (4/3)y) - 5y = 55
Simplify:
(200/3) + (20/3)y - 5y = 55
Multiply all terms by 3 to eliminate the fractions:
200 + 20y - 15y = 165
Combine like terms:
5y = -35
Divide both sides by 5:
y = -7
Now substitute this value of y back into the expression for x:
x = (40/3) + (4/3)(-7)
x = 40/3 - 28/3
x = 12/3
x = 4
The solution to the system of equations is (4, -7).
Therefore, the correct answer is A. (4, -7).
5. To determine how many solutions the system of equations has:
12x - 15y = 18 ...(equation 1)
4x - 5y = 6 ...(equation 2)
One way to determine the number of solutions is by comparing the slopes of the two lines formed by the equations.
Both equations have the same slope of 12/15, which simplifies to 4/5.
When two lines have the same slope and the same y-intercept, they coincide and intersect at infinitely many points, meaning they have an infinite number of solutions.
Therefore, the correct answer is D. infinite solutions.