Solve the following systems of equations using the elimination method.

Question

ScanSolve AI Tutor · Accepted Answer

Here are the solutions for Exercise 4 using the elimination method: 1. (a) Given the equations: 2x + y = 5 (1); x − y = 1 (2) Step 1: Add equation (1) and equation (2) to eliminate y. (2x + y) + (x − y) = 5 + 1; 3x = 6 Step 2: Solve for x. x = 6/3; x = 2; Step 3: Substitute x = 2 into equation (2). 2 − y = 1; y = 2 − 1; y = 1; The solution is x=2, y=1. 1. (b) Given the equations: 2x − y = 5 (1); x + y = 4 (2) Step 1: Add equation (1) and equation (2) to eliminate y. (2x − y) + (x + y) = 5 + 4; 3x = 9 Step 2: Solve for x. x = 9/3; x = 3; Step 3: Substitute x = 3 into equation (2). 3 + y = 4; y = 4 − 3; y = 1; The solution is x=3, y=1. 2. (a) Given the equations: x + 2y = 11 (1); 2x − 2y = 10 (2) Step 1: Add equation (1) and equation (2) to eliminate y. (x + 2y) + (2x − 2y) = 11 + 10; 3x = 21 Step 2: Solve for x. x = 21/3; x = 7; Step 3: Substitute x = 7 into equation (1). 7 + 2y = 11; 2y = 11 − 7; 2y = 4 Step 4: Solve for y. y = 4/2; y = 2; The solution is x=7, y=2. 2. (b) Given the equations: 2a − b = 1 (1); 3a + 2b = 19 (2) Step 1: Multiply equation (1) by 2 to make the coefficient of b equal to −2. 2(2a − b) = 2(1); 4a − 2b = 2 (3) Step 2: Add equation (3) and equation (2) to eliminate b. (4a − 2b) + (3a + 2b) = 2 + 19; 7a = 21 Step 3: Solve for a. a = 21/7; a = 3; Step 4: Substitute a = 3 into equation (1). 2(3) − b = 1; 6 − b = 1; b = 6 − 1; b = 5; The solution is a=3, b=5. 3. (a) Given the equations: 5p + 2q = 18 (1); p + 2q = 2 (2) Step 1: Subtract equation (2) from equation (1) to eliminate q. (5p + 2q) − (p + 2q) = 18 − 2; 4p = 16 Step 2: Solve for p. p = 16/4; p = 4; Step 3: Substitute p = 4 into equation (2). 4 + 2q = 2; 2q = 2 − 4; 2q = −2 Step 4: Solve for q. q = (−2)/(2); q = −1; The solution is p=4, q=−1. 3. (b) Given the equations: 3p + 5k = 30 (1); 2p + 3k = 19 (2) Step 1: Multiply equation (1) by 2 and equation (2) by 3 to make the coefficients of p equal. 2(3p + 5k) = 2(30) ⇒ 6p + 10k = 60 (3); 3(2p + 3k) = 3(19) ⇒ 6p + 9k = 57 (4) Step 2: Subtract equation (4) from equation (3) to eliminate p. (6p + 10k) − (6p + 9k) = 60 − 57; k = 3; Step 3: Substitute k = 3 into equation (1). 3p + 5(3) = 30; 3p + 15 = 30; 3p = 30 − 15; 3p = 15 Step 4: Solve for p. p = 15/3; p = 5; The solution is p=5, k=3. 4. (a) Given the equations: 4m + 3n = 28 (1); 6m + 2n = 32 (2) Step 1: Multiply equation (1) by 3 and equation (2) by 2 to make the coefficients of m equal. 3(4m + 3n) = 3(28) ⇒ 12m + 9n = 84 (3); 2(6m + 2n) = 2(32) ⇒ 12m + 4n = 64 (4) Step 2: Subtract equation (4) from equation (3) to eliminate m. (12m + 9n) − (12m + 4n) = 84 − 64; 5n = 20 Step 3: Solve for n. n = 20/5; n = 4; Step 4: Substitute n = 4 into equation (1). 4m + 3(4) = 28; 4m + 12 = 28; 4m = 28 − 12; 4m = 16 Step 5: Solve for m. m = 16/4; m = 4; The solution is m=4, n=4. 4. (b) Given the equations: 3x + 5y = 7 (1); 4x + 3y = 2 (2) Step 1: Multiply equation (1) by 4 and equation (2) by 3 to make the coefficients of x equal. 4(3x + 5y) = 4(7) ⇒ 12x + 20y = 28 (3); 3(4x + 3y) = 3(2) ⇒ 12x + 9y = 6 (4) Step 2: Subtract equation (4) from equation (3) to eliminate x. (12x + 20y) − (12x + 9y) = 28 − 6; 11y = 22 Step 3: Solve for y. y = 22/11; y = 2; Step 4: Substitute y = 2 into equation (1). 3x + 5(2) = 7; 3x + 10 = 7; 3x = 7 − 10; 3x = −3 Step 5: Solve for x. x = (−3)/(3); x = −1; The solution is x=−1, y=2.

Solve the following systems of equations using the elimination method.

Step-by-step solution

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

Solve for $x$ .

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

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

Solve for $x$ .

Substitute $x = 3$ into equation (2).

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

Solve for $x$ .

Substitute $x = 7$ into equation (1).

Solve for $y$ .

Multiply equation (1) by 2 to make the coefficient of $b$ equal to $-2$ .

Add equation (3) and equation (2) to eliminate $b$ .

Solve for $a$ .

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

Subtract equation (2) from equation (1) to eliminate $q$ .

Solve for $p$ .

Substitute $p = 4$ into equation (2).

Solve for $q$ .

Multiply equation (1) by 2 and equation (2) by 3 to make the coefficients of $p$ equal.

Subtract equation (4) from equation (3) to eliminate $p$ .

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

Solve for $p$ .

Multiply equation (1) by 3 and equation (2) by 2 to make the coefficients of $m$ equal.

Subtract equation (4) from equation (3) to eliminate $m$ .

Solve for $n$ .

Substitute $n = 4$ into equation (1).

Solve for $m$ .

Multiply equation (1) by 4 and equation (2) by 3 to make the coefficients of $x$ equal.

Subtract equation (4) from equation (3) to eliminate $x$ .

Solve for $y$ .

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

Solve for $x$ .

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More Mathematics Questions

Solve the following systems of equations using the elimination method.

Step-by-step solution

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

Solve for xxx.

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

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

Solve for xxx.

Substitute x=3x = 3x=3 into equation (2).

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

Solve for xxx.

Substitute x=7x = 7x=7 into equation (1).

Solve for yyy.

Multiply equation (1) by 2 to make the coefficient of bbb equal to −2-2−2.

Add equation (3) and equation (2) to eliminate bbb.

Solve for aaa.

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

Subtract equation (2) from equation (1) to eliminate qqq.

Solve for ppp.

Substitute p=4p = 4p=4 into equation (2).

Solve for qqq.

Multiply equation (1) by 2 and equation (2) by 3 to make the coefficients of ppp equal.

Subtract equation (4) from equation (3) to eliminate ppp.

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

Solve for ppp.

Multiply equation (1) by 3 and equation (2) by 2 to make the coefficients of mmm equal.

Subtract equation (4) from equation (3) to eliminate mmm.

Solve for nnn.

Substitute n=4n = 4n=4 into equation (1).

Solve for mmm.

Multiply equation (1) by 4 and equation (2) by 3 to make the coefficients of xxx equal.

Subtract equation (4) from equation (3) to eliminate xxx.

Solve for yyy.

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

Solve for xxx.

Still stuck on this one?

More Mathematics Questions

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

Solve for $x$ .

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

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

Solve for $x$ .

Substitute $x = 3$ into equation (2).

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

Solve for $x$ .

Substitute $x = 7$ into equation (1).

Solve for $y$ .

Multiply equation (1) by 2 to make the coefficient of $b$ equal to $-2$ .

Add equation (3) and equation (2) to eliminate $b$ .

Solve for $a$ .

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

Subtract equation (2) from equation (1) to eliminate $q$ .

Solve for $p$ .

Substitute $p = 4$ into equation (2).

Solve for $q$ .

Multiply equation (1) by 2 and equation (2) by 3 to make the coefficients of $p$ equal.

Subtract equation (4) from equation (3) to eliminate $p$ .

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

Solve for $p$ .

Multiply equation (1) by 3 and equation (2) by 2 to make the coefficients of $m$ equal.

Subtract equation (4) from equation (3) to eliminate $m$ .

Solve for $n$ .

Substitute $n = 4$ into equation (1).

Solve for $m$ .

Multiply equation (1) by 4 and equation (2) by 3 to make the coefficients of $x$ equal.

Subtract equation (4) from equation (3) to eliminate $x$ .

Solve for $y$ .

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

Solve for $x$ .