Solve each system of equations using any of the following methods: substitution,
elimination, echelon, or Gauss-Jordan. If there are no solutions, say so. If there are an
infinite number of solutions, parametrize the answer using y as the parameter.
(a) x − 4y = −15
−2x + y = −19
(b) 4x − 10y = 42
−2x + 5y = 17
(c) 15x + 9y = 27
20x + 12y = 36
(d) 1
2
x + 2y = 2
−2x +
3
4
y = 1
(a) Using substitution, we can solve for one variable in terms of the other and substitute into the other equation. For example, from the first equation, we have x = 4y - 15. Substituting this into the second equation, we get -2(4y-15) + y = -19, which simplifies to -8y + 30 + y = -19, or -7y = -49, or y = 7. Substituting this back into the first equation, we get x = 4(7) - 15 = 13. Therefore, the solution is (13, 7).
(b) Using elimination, we can multiply the second equation by 2 and add it to the first equation to eliminate x. We get 4x - 10y + (-4x + 10y) = 42 + 2(17), simplifying to 0 = 76, which is false. Therefore, there is no solution.
(c) Using elimination, we can multiply the first equation by -4 and add it to the second equation to eliminate x. We get 20x + 12y + (-20x + 36y) = 36 - 108, simplifying to 0 = -72. This is also false. Therefore, there is no solution.
(d) Using substitution, we can solve for x in terms of y from the second equation. We get x = (2/3)y + (2/3). Substituting this into the first equation, we get (1/2)((2/3)y + (2/3)) + 2y = 2, which simplifies to (1/3)y + (1/3) + 2y = 2, or (7/3)y = 5/3, or y = (5/3)(3/7) = 5/7. Substituting this back into the expression for x, we get x = (2/3)(5/7) + (2/3) = 4/7. Therefore, the solution is (4/7, 5/7).
(a) To solve the system of equations using the substitution method:
Step 1: Solve one equation for one variable in terms of the other variable.
From the second equation: -2x + y = -19
Rearrange the equation to get:
y = 2x - 19
Step 2: Substitute the expression for the variable found in step 1 into the other equation.
x - 4y = -15
Substitute y with 2x - 19:
x - 4(2x - 19) = -15
Simplify and solve for x:
x - 8x + 76 = -15
-7x = -15 - 76
-7x = -91
x = -91/-7
x = 13
Step 3: Substitute the value of x found in step 2 into one of the original equations to solve for y.
Using the first equation: x - 4y = -15
Substitute x with 13:
13 - 4y = -15
-4y = -15 - 13
-4y = -28
y = -28 / -4
y = 7
Therefore, the solutions to the system of equations are x = 13 and y = 7.
(b) To solve the system of equations using the elimination method:
Step 1: Multiply both equations by suitable constants to create opposite coefficients for one of the variables.
Multiply the first equation by 2: 8x - 20y = 84
Multiply the second equation by 4: -8x + 20y = 68
Step 2: Add the two equations together to eliminate one variable.
(8x - 20y) + (-8x + 20y) = 84 + 68
0 = 152
Since we obtain a contradiction (0 = 152), the system of equations has no solution.
(c) To solve the system of equations using the echelon or Gauss-Jordan method:
Step 1: Write the augmented matrix for the system of equations.
[ 15 9 | 27 ]
[ 20 12 | 36 ]
Step 2: Perform row operations to reduce the matrix to echelon form or row-reduced echelon form.
First, subtract 4 times the first row from the second row:
[ 15 9 | 27 ]
[ 0 -24 | -36 ]
Next, multiply the second row by -1/24 to make the leading coefficient of the second row equal to 1:
[ 15 9 | 27 ]
[ 0 1 | 3/2 ]
Lastly, subtract 9 times the second row from the first row:
[ 15 0 | 0 ]
[ 0 1 | 3/2 ]
Step 3: Determine the values of x and y.
From the modified matrix we have:
15x = 0 → x = 0
y = 3/2
Therefore, the solution to the system of equations is x = 0 and y = 3/2.
(d) To solve the system of equations using the substitution method:
Step 1: Solve the second equation for one variable in terms of the other variable.
From the second equation: -2x + (3/4)y = 1
Rearrange the equation to get:
-2x = - (3/4)y + 1
Divide by -2:
x = (3/8)y - 1/2
Step 2: Substitute the expression for x found in step 1 into the first equation.
(1/2)x + 2y = 2
Substitute x with (3/8)y - 1/2:
(1/2)((3/8)y - 1/2) + 2y = 2
Simplify and solve for y:
(3/16)y - 1/4 + 2y = 2
(3/16)y + 2(16/16)y = 2 + 4/4
(19/16)y = 10/4
y = (10/4) / (19/16)
y = 10/4 * 16/19
y = 80/76
y = 10/9
Step 3: Substitute the value of y found in step 2 into one of the original equations to solve for x.
Using the second equation: -2x + (3/4)y = 1
Substitute y with 10/9:
-2x + (3/4)(10/9) = 1
-2x + 5/6 = 1
-2x = 1 - 5/6
-12x = 6 - 5
-12x = 1
x = 1/-12
x = -1/12
Therefore, the solution to the system of equations is x = -1/12 and y = 10/9.