Using the elimination method, solve each of the following pairs of simultaneous equations

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Using the elimination method, solve each of the following pairs of simultaneous equations. (a) 4x – y – 7 = 0 4x + 3y – 11 = 0 (c) 5x – 3y – 2 = 0 x + 5y – 6 = 0 (e) 7x + 3y – 8 = 0 3x – 4y – 14 = 0 (b) 7x + 2y – 33 = 0 3y – 7x – 17 = 0 (d) 5x – 3y – 13 = 0 7x – 6y – 20 = 0 (f) 3x + 5y + 8 = 0 4x + 13y – 2 = 0

The Correct Answer and Explanation is:

Let’s solve the pairs of simultaneous equations using the elimination method. This method involves manipulating the equations to eliminate one of the variables, making it easier to solve for the other.

(a) 4x – y – 7 = 0

CopyEdit  4x + 3y - 11 = 0

Step 1: Rearrange the equations into standard form:

  1. 4x−y=74x – y = 74x−y=7 (Equation 1)
  2. 4x+3y=114x + 3y = 114x+3y=11 (Equation 2)

Step 2: Subtract Equation 1 from Equation 2 to eliminate xxx:(4x+3y)−(4x−y)=11−7(4x + 3y) – (4x – y) = 11 – 7(4x+3y)−(4x−y)=11−7

This simplifies to:4x+3y−4x+y=4⇒4y=4⇒y=14x + 3y – 4x + y = 4 \quad \Rightarrow \quad 4y = 4 \quad \Rightarrow \quad y = 14x+3y−4x+y=4⇒4y=4⇒y=1

Step 3: Substitute y=1y = 1y=1 into Equation 1:4x−1=7⇒4x=8⇒x=24x – 1 = 7 \quad \Rightarrow \quad 4x = 8 \quad \Rightarrow \quad x = 24x−1=7⇒4x=8⇒x=2

Solution for (a):x=2, y=1x = 2, \, y = 1x=2,y=1

(b) 7x + 2y – 33 = 0

CopyEdit  3y - 7x - 17 = 0

Step 1: Rearrange the equations:

  1. 7x+2y=337x + 2y = 337x+2y=33 (Equation 1)
  2. −7x+3y=17-7x + 3y = 17−7x+3y=17 (Equation 2)

Step 2: Add the two equations to eliminate xxx:(7x+2y)+(−7x+3y)=33+17(7x + 2y) + (-7x + 3y) = 33 + 17(7x+2y)+(−7x+3y)=33+17

This simplifies to:5y=50⇒y=105y = 50 \quad \Rightarrow \quad y = 105y=50⇒y=10

Step 3: Substitute y=10y = 10y=10 into Equation 1:7x+2(10)=33⇒7x+20=33⇒7x=13⇒x=1377x + 2(10) = 33 \quad \Rightarrow \quad 7x + 20 = 33 \quad \Rightarrow \quad 7x = 13 \quad \Rightarrow \quad x = \frac{13}{7}7x+2(10)=33⇒7x+20=33⇒7x=13⇒x=713​

Solution for (b):x=137, y=10x = \frac{13}{7}, \, y = 10x=713​,y=10

(c) 5x – 3y – 2 = 0

nginxCopyEdit  x + 5y - 6 = 0

Step 1: Rearrange the equations:

  1. 5x−3y=25x – 3y = 25x−3y=2 (Equation 1)
  2. x+5y=6x + 5y = 6x+5y=6 (Equation 2)

Step 2: Multiply Equation 2 by 5 to align the coefficients of xxx:5(x+5y)=5(6)5(x + 5y) = 5(6)5(x+5y)=5(6)

This gives:5x+25y=30(Equation 3)5x + 25y = 30 \quad \text{(Equation 3)}5x+25y=30(Equation 3)

Step 3: Subtract Equation 1 from Equation 3 to eliminate xxx:(5x+25y)−(5x−3y)=30−2(5x + 25y) – (5x – 3y) = 30 – 2(5x+25y)−(5x−3y)=30−2

This simplifies to:28y=28⇒y=128y = 28 \quad \Rightarrow \quad y = 128y=28⇒y=1

Step 4: Substitute y=1y = 1y=1 into Equation 2:x+5(1)=6⇒x+5=6⇒x=1x + 5(1) = 6 \quad \Rightarrow \quad x + 5 = 6 \quad \Rightarrow \quad x = 1x+5(1)=6⇒x+5=6⇒x=1

Solution for (c):x=1, y=1x = 1, \, y = 1x=1,y=1

(d) 5x – 3y – 13 = 0

CopyEdit  7x - 6y - 20 = 0

Step 1: Rearrange the equations:

  1. 5x−3y=135x – 3y = 135x−3y=13 (Equation 1)
  2. 7x−6y=207x – 6y = 207x−6y=20 (Equation 2)

Step 2: Multiply Equation 1 by 2 to align the coefficients of yyy:2(5x−3y)=2(13)2(5x – 3y) = 2(13)2(5x−3y)=2(13)

This gives:10x−6y=26(Equation 3)10x – 6y = 26 \quad \text{(Equation 3)}10x−6y=26(Equation 3)

Step 3: Subtract Equation 2 from Equation 3 to eliminate yyy:(10x−6y)−(7x−6y)=26−20(10x – 6y) – (7x – 6y) = 26 – 20(10x−6y)−(7x−6y)=26−20

This simplifies to:3x=6⇒x=23x = 6 \quad \Rightarrow \quad x = 23x=6⇒x=2

Step 4: Substitute x=2x = 2x=2 into Equation 1:5(2)−3y=13⇒10−3y=13⇒−3y=3⇒y=−15(2) – 3y = 13 \quad \Rightarrow \quad 10 – 3y = 13 \quad \Rightarrow \quad -3y = 3 \quad \Rightarrow \quad y = -15(2)−3y=13⇒10−3y=13⇒−3y=3⇒y=−1

Solution for (d):x=2, y=−1x = 2, \, y = -1x=2,y=−1

(e) 7x + 3y – 8 = 0

CopyEdit  3x - 4y - 14 = 0

Step 1: Rearrange the equations:

  1. 7x+3y=87x + 3y = 87x+3y=8 (Equation 1)
  2. 3x−4y=143x – 4y = 143x−4y=14 (Equation 2)

Step 2: Multiply Equation 1 by 3 and Equation 2 by 7 to align the coefficients of xxx:3(7x+3y)=3(8)⇒21x+9y=24(Equation 3)3(7x + 3y) = 3(8) \quad \Rightarrow \quad 21x + 9y = 24 \quad \text{(Equation 3)}3(7x+3y)=3(8)⇒21x+9y=24(Equation 3)7(3x−4y)=7(14)⇒21x−28y=98(Equation 4)7(3x – 4y) = 7(14) \quad \Rightarrow \quad 21x – 28y = 98 \quad \text{(Equation 4)}7(3x−4y)=7(14)⇒21x−28y=98(Equation 4)

Step 3: Subtract Equation 3 from Equation 4 to eliminate xxx:(21x−28y)−(21x+9y)=98−24(21x – 28y) – (21x + 9y) = 98 – 24(21x−28y)−(21x+9y)=98−24

This simplifies to:−37y=74⇒y=−2-37y = 74 \quad \Rightarrow \quad y = -2−37y=74⇒y=−2

Step 4: Substitute y=−2y = -2y=−2 into Equation 1:7x+3(−2)=8⇒7x−6=8⇒7x=14⇒x=27x + 3(-2) = 8 \quad \Rightarrow \quad 7x – 6 = 8 \quad \Rightarrow \quad 7x = 14 \quad \Rightarrow \quad x = 27x+3(−2)=8⇒7x−6=8⇒7x=14⇒x=2

Solution for (e):x=2, y=−2x = 2, \, y = -2x=2,y=−2

(f) 3x + 5y + 8 = 0

CopyEdit  4x + 13y - 2 = 0

Step 1: Rearrange the equations:

  1. 3x+5y=−83x + 5y = -83x+5y=−8 (Equation 1)
  2. 4x+13y=24x + 13y = 24x+13y=2 (Equation 2)

Step 2: Multiply Equation 1 by 4 and Equation 2 by 3 to align the coefficients of xxx:4(3x+5y)=4(−8)⇒12x+20y=−32(Equation 3)4(3x + 5y) = 4(-8) \quad \Rightarrow \quad 12x + 20y = -32 \quad \text{(Equation 3)}4(3x+5y)=4(−8)⇒12x+20y=−32(Equation 3)3(4x+13y)=3(2)⇒12x+39y=6(Equation 4)3(4x + 13y) = 3(2) \quad \Rightarrow \quad 12x + 39y = 6 \quad \text{(Equation 4)}3(4x+13y)=3(2)⇒12x+39y=6(Equation 4)

Step 3: Subtract Equation 3 from Equation 4 to eliminate xxx:(12x+39y)−(12x+20y)=6−(−32)(12x + 39y) – (12x + 20y) = 6 – (-32)(12x+39y)−(12x+20y)=6−(−32)

This simplifies to:19y=38⇒y=219y = 38 \quad \Rightarrow \quad y = 219y=38⇒y=2

Step 4: Substitute y=2y = 2y=2 into Equation 1:3x+5(2)=−8⇒3x+10=−8⇒3x=−18⇒x=−63x + 5(2) = -8 \quad \Rightarrow \quad 3x + 10 = -8 \quad \Rightarrow \quad 3x = -18 \quad \Rightarrow \quad x = -63x+5(2)=−8⇒3x+10=−8⇒3x=−18⇒x=−6

Solution for (f):x=−6, y=2x = -6, \, y = 2x=−6,y=2

Summary of Solutions:

  • (a) x=2, y=1x = 2, \, y = 1x=2,y=1
  • (b) x=137, y=10x = \frac{13}{7}, \, y = 10x=713​,y=10
  • (c) x=1, y=1x = 1, \, y = 1x=1,y=1
  • (d) x=2, y=−1x = 2, \, y = -1x=2,y=−1
  • (e) x=2, y=−2x = 2, \, y = -2x=2,y=−2
  • (f) x=−6, y=2x = -6, \, y = 2x=−6,y=2

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