Answer:
Let's solve the given systems of equations using both the Elimination Method and the Substitution Method.
**1. x + 2y = 1
2x - y = 2**
**Elimination Method:**
Step 1: Multiply both sides of the second equation by 2 to make the coefficients of y the same in both equations.
2(2x - y) = 2(2)
4x - 2y = 4
Step 2: Now, subtract the first equation from the modified second equation to eliminate y.
(4x - 2y) - (x + 2y) = 4 - 1
4x - 2y - x - 2y = 3
3x - 4y = 3
Step 3: Now, solve for x or y in one of the equations. Let's use the first equation x + 2y = 1 to solve for x:
x = 1 - 2y
Step 4: Substitute the value of x into the equation 3x - 4y = 3:
3(1 - 2y) - 4y = 3
3 - 6y - 4y = 3
-10y = 0
y = 0
Step 5: Now that we have the value of y, substitute it back into the equation x + 2y = 1 to find x:
x + 2(0) = 1
x = 1
So, the solution to the system of equations is x = 1 and y = 0.
**Substitution Method:**
Step 1: Solve one of the equations for either x or y. Let's use the first equation x + 2y = 1 to solve for x:
Step 2: Substitute the value of x into the second equation 2x - y = 2:
2(1 - 2y) - y = 2
2 - 4y - y = 2
-5y = 0
Step 3: Now that we have the value of y, substitute it back into x + 2y = 1 to find x:
**2. 3x - 2y = 0
x + y = 3**
2(x + y) = 2(3)
2x + 2y = 6
Step 2: Now, add the first equation to the modified second equation to eliminate y.
3x - 2y + 2x + 2y = 0 + 6
5x = 6
x = 6/5
Step 3: Now that we have the value of x, substitute it back into the second equation x + y = 3 to find y:
6/5 + y = 3
y = 3 - 6/5
y = 15/5 - 6/5
y = 9/5
So, the solution to the system of equations is x = 6/5 and y = 9/5.
Step 1: Solve one of the equations for either x or y. Let's use the second equation x + y = 3 to solve for y:
y = 3 - x
Step 2: Substitute the value of y into the first equation 3x - 2y = 0:
3x - 2(3 - x) = 0
3x - 6 + 2x = 0
5x - 6 = 0
Step 3: Now that we have the value of x, substitute it back into y = 3 - x to find y:
Both methods yield the same solution for each system of equations.
Let's solve the given systems of equations using the Elimination Method and the Substitution Method:
1. x + 2y = 1
2x - y = 2
Step 1: Multiply the first equation by 2 to eliminate the y variable:
2(x + 2y) = 2(1)
2x + 4y = 2
Step 2: Now, add the second equation to the modified first equation:
(2x + 4y) + (2x - y) = 2 + 2
4x + 3y = 4
Step 3: Now, solve for y:
3y = 4 - 4x
y = (4 - 4x) / 3
Step 4: Substitute the value of y into the first equation and solve for x:
x + 2((4 - 4x) / 3) = 1
x + (8 - 8x) / 3 = 1
Step 5: Now, multiply the whole equation by 3 to eliminate the fraction:
3x + 8 - 8x = 3
-5x + 8 = 3
Step 6: Now, isolate x:
-5x = 3 - 8
-5x = -5
x = -5 / -5
Step 7: Substitute the value of x into either equation to solve for y:
2(1) - y = 2
2 - y = 2
-y = 2 - 2
-y = 0
Step 1: Solve one equation for one variable in terms of the other variable.
From the second equation, solve for y:
y = 2x - 2
Step 2: Substitute the value of y into the first equation:
x + 2(2x - 2) = 1
x + 4x - 4 = 1
Step 3: Combine like terms:
5x - 4 = 1
Step 4: Isolate x:
5x = 1 + 4
5x = 5
x = 5 / 5
Step 5: Now, substitute the value of x back into one of the original equations to solve for y:
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2. 3x - 2y = 0
x + y = 3
Step 1: Multiply the second equation by 2 to make the coefficients of y in both equations the same:
Step 2: Now, add the first equation to the modified second equation:
(3x - 2y) + (2x + 2y) = 0 + 6
Step 3: Isolate x:
x = 6 / 5
Step 4: Substitute the value of x into the second equation and solve for y:
(6 / 5) + y = 3
Step 5: Solve for y:
Step 6: Convert 3 to a fraction with the denominator 5:
y = (15/5) - (6/5)
y = (15 - 6) / 5
y = 9 / 5
Step 3: Simplify the equation:
Step 4: Combine like terms:
Step 5: Isolate x:
Step 6: Now, substitute the value of x back into one of the original equations to solve for y:
Step 7: Solve for y:
Step 8: Convert 3 to a fraction with the denominator 5:
Step-by-step explanation:
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Answers & Comments
Answer:
Let's solve the given systems of equations using both the Elimination Method and the Substitution Method.
**1. x + 2y = 1
2x - y = 2**
**Elimination Method:**
Step 1: Multiply both sides of the second equation by 2 to make the coefficients of y the same in both equations.
2(2x - y) = 2(2)
4x - 2y = 4
Step 2: Now, subtract the first equation from the modified second equation to eliminate y.
(4x - 2y) - (x + 2y) = 4 - 1
4x - 2y - x - 2y = 3
3x - 4y = 3
Step 3: Now, solve for x or y in one of the equations. Let's use the first equation x + 2y = 1 to solve for x:
x = 1 - 2y
Step 4: Substitute the value of x into the equation 3x - 4y = 3:
3(1 - 2y) - 4y = 3
3 - 6y - 4y = 3
-10y = 0
y = 0
Step 5: Now that we have the value of y, substitute it back into the equation x + 2y = 1 to find x:
x + 2(0) = 1
x = 1
So, the solution to the system of equations is x = 1 and y = 0.
**Substitution Method:**
Step 1: Solve one of the equations for either x or y. Let's use the first equation x + 2y = 1 to solve for x:
x = 1 - 2y
Step 2: Substitute the value of x into the second equation 2x - y = 2:
2(1 - 2y) - y = 2
2 - 4y - y = 2
-5y = 0
y = 0
Step 3: Now that we have the value of y, substitute it back into x + 2y = 1 to find x:
x + 2(0) = 1
x = 1
So, the solution to the system of equations is x = 1 and y = 0.
**2. 3x - 2y = 0
x + y = 3**
**Elimination Method:**
Step 1: Multiply both sides of the second equation by 2 to make the coefficients of y the same in both equations.
2(x + y) = 2(3)
2x + 2y = 6
Step 2: Now, add the first equation to the modified second equation to eliminate y.
3x - 2y + 2x + 2y = 0 + 6
5x = 6
x = 6/5
Step 3: Now that we have the value of x, substitute it back into the second equation x + y = 3 to find y:
6/5 + y = 3
y = 3 - 6/5
y = 15/5 - 6/5
y = 9/5
So, the solution to the system of equations is x = 6/5 and y = 9/5.
**Substitution Method:**
Step 1: Solve one of the equations for either x or y. Let's use the second equation x + y = 3 to solve for y:
y = 3 - x
Step 2: Substitute the value of y into the first equation 3x - 2y = 0:
3x - 2(3 - x) = 0
3x - 6 + 2x = 0
5x - 6 = 0
5x = 6
x = 6/5
Step 3: Now that we have the value of x, substitute it back into y = 3 - x to find y:
y = 3 - 6/5
y = 15/5 - 6/5
y = 9/5
So, the solution to the system of equations is x = 6/5 and y = 9/5.
Both methods yield the same solution for each system of equations.
Answer:
Let's solve the given systems of equations using the Elimination Method and the Substitution Method:
1. x + 2y = 1
2x - y = 2
**Elimination Method:**
Step 1: Multiply the first equation by 2 to eliminate the y variable:
2(x + 2y) = 2(1)
2x + 4y = 2
Step 2: Now, add the second equation to the modified first equation:
(2x + 4y) + (2x - y) = 2 + 2
4x + 3y = 4
Step 3: Now, solve for y:
3y = 4 - 4x
y = (4 - 4x) / 3
Step 4: Substitute the value of y into the first equation and solve for x:
x + 2((4 - 4x) / 3) = 1
x + (8 - 8x) / 3 = 1
Step 5: Now, multiply the whole equation by 3 to eliminate the fraction:
3x + 8 - 8x = 3
-5x + 8 = 3
Step 6: Now, isolate x:
-5x = 3 - 8
-5x = -5
x = -5 / -5
x = 1
Step 7: Substitute the value of x into either equation to solve for y:
2(1) - y = 2
2 - y = 2
-y = 2 - 2
-y = 0
y = 0
So, the solution to the system of equations is x = 1 and y = 0.
**Substitution Method:**
Step 1: Solve one equation for one variable in terms of the other variable.
From the second equation, solve for y:
y = 2x - 2
Step 2: Substitute the value of y into the first equation:
x + 2(2x - 2) = 1
x + 4x - 4 = 1
Step 3: Combine like terms:
5x - 4 = 1
Step 4: Isolate x:
5x = 1 + 4
5x = 5
x = 5 / 5
x = 1
Step 5: Now, substitute the value of x back into one of the original equations to solve for y:
2(1) - y = 2
2 - y = 2
-y = 2 - 2
-y = 0
y = 0
So, the solution to the system of equations is x = 1 and y = 0.
---
2. 3x - 2y = 0
x + y = 3
**Elimination Method:**
Step 1: Multiply the second equation by 2 to make the coefficients of y in both equations the same:
2(x + y) = 2(3)
2x + 2y = 6
Step 2: Now, add the first equation to the modified second equation:
(3x - 2y) + (2x + 2y) = 0 + 6
5x = 6
Step 3: Isolate x:
x = 6 / 5
Step 4: Substitute the value of x into the second equation and solve for y:
(6 / 5) + y = 3
Step 5: Solve for y:
y = 3 - 6/5
Step 6: Convert 3 to a fraction with the denominator 5:
y = (15/5) - (6/5)
y = (15 - 6) / 5
y = 9 / 5
So, the solution to the system of equations is x = 6/5 and y = 9/5.
**Substitution Method:**
Step 1: Solve one equation for one variable in terms of the other variable.
From the second equation, solve for y:
y = 3 - x
Step 2: Substitute the value of y into the first equation:
3x - 2(3 - x) = 0
Step 3: Simplify the equation:
3x - 6 + 2x = 0
Step 4: Combine like terms:
5x - 6 = 0
Step 5: Isolate x:
5x = 6
x = 6 / 5
Step 6: Now, substitute the value of x back into one of the original equations to solve for y:
(6 / 5) + y = 3
Step 7: Solve for y:
y = 3 - 6/5
Step 8: Convert 3 to a fraction with the denominator 5:
y = (15/5) - (6/5)
y = (15 - 6) / 5
y = 9 / 5
So, the solution to the system of equations is x = 6/5 and y = 9/5.
Step-by-step explanation: