Solve the following systems of linear equations using the elimination method: a) x + 3y = 8, 5x + 7y = 24 b)…

Question

Solve the following systems of linear equations using the elimination method: a) x + 3y = 8, 5x + 7y = 24 b) 5x + 4y = 22, 4x + 5y = 23 c) 5x − 3y = 22, 10x + 7y = 18

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Here are the solutions for each system of equations using the elimination method: a) We have the system: x + 3y = 8 (1); 5x + 7y = 24 (2) Step 1: Multiply equation (1) by 5 to eliminate x. 5(x + 3y) = 5(8); 5x + 15y = 40 (3) Step 2: Subtract equation (2) from equation (3). (5x + 15y) − (5x + 7y) = 40 − 24; 8y = 16; y = 2; Step 3: Substitute y = 2 into equation (1). x + 3(2) = 8; x + 6 = 8; x = 2; The solution is x=2, y=2. b) We have the system: 5x + 4y = 22 (1); 4x + 5y = 23 (2) Step 1: Multiply equation (1) by 4 and equation (2) by 5 to eliminate x. 4(5x + 4y) = 4(22) ⇒ 20x + 16y = 88 (3); 5(4x + 5y) = 5(23) ⇒ 20x + 25y = 115 (4) Step 2: Subtract equation (3) from equation (4). (20x + 25y) − (20x + 16y) = 115 − 88; 9y = 27; y = 3; Step 3: Substitute y = 3 into equation (1). 5x + 4(3) = 22; 5x + 12 = 22; 5x = 10; x = 2; The solution is x=2, y=3. c) We have the system: 5x − 3y = 22 (1); 10x + 7y = 18 (2) Step 1: Multiply equation (1) by 2 to eliminate x. 2(5x − 3y) = 2(22); 10x − 6y = 44 (3) Step 2: Subtract equation (3) from equation (2). (10x + 7y) − (10x − 6y) = 18 − 44; 13y = −26; y = −2; Step 3: Substitute y = −2 into equation (1). 5x − 3(−2) = 22; 5x + 6 = 22; 5x = 16; x = 16/5; The solution is x=16/5, y=−2. d) We have the system: x + y = 3 (1); 3x − y = 5 (2) Step 1: Add equation (1) and equation (2) to eliminate y. (x + y) + (3x − y) = 3 + 5; 4x = 8; x = 2; Step 2: Substitute x = 2 into equation (1). 2 + y = 3; y = 1; The solution is x=2, y=1. e) We have the system: 3x + y = 5 (1); 2x − y = 0 (2) Step 1: Add equation (1) and equation (2) to eliminate y. (3x + y) + (2x − y) = 5 + 0; 5x = 5; x = 1; Step 2: Substitute x = 1 into equation (2). 2(1) − y = 0; 2 − y = 0; y = 2; The solution is x=1, y=2. f) We have the system: x + 4y = 6 (1); 2x + 5y = 6 (2) Step 1: Multiply equation (1) by 2 to eliminate x. 2(x + 4y) = 2(6); 2x + 8y = 12 (3) Step 2: Subtract equation (2) from equation (3). (2x + 8y) − (2x + 5y) = 12 − 6; 3y = 6; y = 2; Step 3: Substitute y = 2 into equation (1). x + 4(2) = 6; x + 8 = 6; x = −2; The solution is x=−2, y=2.

Solve the following systems of linear equations using the elimination method: a) x + 3y = 8, 5x + 7y = 24 b) 5x + 4y = 22, 4x + 5y = 23 c) 5x − 3y = 22, 10x + 7y = 18

Answers

Step-by-step solution

Part a· 3 steps

Multiply equation (1) by 5 to eliminate $x$ .

Subtract equation (2) from equation (3).

Substitute $y = 2$ into equation (1).

Part b· 3 steps
$x=2, y=3$

Multiply equation (1) by 4 and equation (2) by 5 to eliminate $x$ .

Subtract equation (3) from equation (4).

Substitute $y = 3$ into equation (1).

Part c· 3 steps
$x=\frac{16}{5}, y=-2$

Multiply equation (1) by 2 to eliminate $x$ .

Subtract equation (3) from equation (2).

Substitute $y = -2$ into equation (1).

Part d· 2 steps
$x=2, y=1$

Add equation (1) and equation (2) to eliminate $y$ .

Substitute $x = 2$ into equation (1).

Part e· 2 steps
$x=1, y=2$

Add equation (1) and equation (2) to eliminate $y$ .

Substitute $x = 1$ into equation (2).

Part f· 3 steps
$x=-2, y=2$

Multiply equation (1) by 2 to eliminate $x$ .

Subtract equation (2) from equation (3).

Substitute $y = 2$ into equation (1).

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More Mathematics Questions

Solve the following systems of linear equations using the elimination method: a) x + 3y = 8, 5x + 7y = 24 b) 5x + 4y = 22, 4x + 5y = 23 c) 5x − 3y = 22, 10x + 7y = 18

Answers

Step-by-step solution

Part a· 3 steps

Multiply equation (1) by 5 to eliminate xxx.

Subtract equation (2) from equation (3).

Substitute y=2y = 2y=2 into equation (1).

Part b· 3 stepsx=2,y=3x=2, y=3x=2,y=3

Multiply equation (1) by 4 and equation (2) by 5 to eliminate xxx.

Subtract equation (3) from equation (4).

Substitute y=3y = 3y=3 into equation (1).

Part c· 3 stepsx=165,y=−2x=\frac{16}{5}, y=-2x=516​,y=−2

Multiply equation (1) by 2 to eliminate xxx.

Subtract equation (3) from equation (2).

Substitute y=−2y = -2y=−2 into equation (1).

Part d· 2 stepsx=2,y=1x=2, y=1x=2,y=1

Add equation (1) and equation (2) to eliminate yyy.

Substitute x=2x = 2x=2 into equation (1).

Part e· 2 stepsx=1,y=2x=1, y=2x=1,y=2

Add equation (1) and equation (2) to eliminate yyy.

Substitute x=1x = 1x=1 into equation (2).

Part f· 3 stepsx=−2,y=2x=-2, y=2x=−2,y=2

Multiply equation (1) by 2 to eliminate xxx.

Subtract equation (2) from equation (3).

Substitute y=2y = 2y=2 into equation (1).

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More Mathematics Questions

Multiply equation (1) by 5 to eliminate $x$ .

Substitute $y = 2$ into equation (1).

Part b· 3 steps
$x=2, y=3$

Multiply equation (1) by 4 and equation (2) by 5 to eliminate $x$ .

Substitute $y = 3$ into equation (1).

Part c· 3 steps
$x=\frac{16}{5}, y=-2$

Multiply equation (1) by 2 to eliminate $x$ .

Substitute $y = -2$ into equation (1).

Part d· 2 steps
$x=2, y=1$

Add equation (1) and equation (2) to eliminate $y$ .

Substitute $x = 2$ into equation (1).

Part e· 2 steps
$x=1, y=2$

Add equation (1) and equation (2) to eliminate $y$ .

Substitute $x = 1$ into equation (2).

Part f· 3 steps
$x=-2, y=2$

Multiply equation (1) by 2 to eliminate $x$ .

Substitute $y = 2$ into equation (1).