Math / Algebra

QuestionFirm Unt 3. Cesson t2)
2. Tyler is solving this system of equations: $\left\{\begin{array}{l}4 p+2 q=62 \\ 8 p-q=59\end{array}\right.$
They can think of two ways to eliminate a variable and solve the system: - Multiply $4 p+2 q=62$ by 2 , then subtract $8 p-q=59$ from the result. - Multiply $8 p-q=59$ by 2 , then add the result to $4 p+2 q=62$ .
Do both strategies work for solving the system? Explain or show your reasoning.

Studdy Solution

STEP 1

1. We have a system of linear equations with two variables $p$ and $q$ .
2. We are given two methods to eliminate a variable and solve the system.
3. We need to determine if both methods correctly solve the system.

STEP 2

1. Apply the first method: Multiply the first equation by 2 and subtract the second equation.
2. Apply the second method: Multiply the second equation by 2 and add it to the first equation.
3. Verify if both methods lead to a valid solution for the system.

STEP 3

Multiply the first equation $4p + 2q = 62$ by 2:
$2(4p + 2q) = 2 \times 62$
$8p + 4q = 124$

STEP 4

Subtract the second equation $8p - q = 59$ from the result:
$(8p + 4q) - (8p - q) = 124 - 59$
$8p + 4q - 8p + q = 65$
$5q = 65$

STEP 5

Solve for $q$ :
$q = \frac{65}{5} = 13$

STEP 6

Multiply the second equation $8p - q = 59$ by 2:
$2(8p - q) = 2 \times 59$
$16p - 2q = 118$

STEP 7

Add the result to the first equation $4p + 2q = 62$ :
$(16p - 2q) + (4p + 2q) = 118 + 62$
$16p - 2q + 4p + 2q = 180$
$20p = 180$

STEP 8

Solve for $p$ :
$p = \frac{180}{20} = 9$

STEP 9

Verify the solution by substituting $p = 9$ and $q = 13$ into the original equations:
First equation: $4p + 2q = 62$
$4(9) + 2(13) = 36 + 26 = 62$
Second equation: $8p - q = 59$
$8(9) - 13 = 72 - 13 = 59$
Both equations are satisfied, confirming the solution is correct.
Both strategies work for solving the system, and the solution is:
$p = 9, \, q = 13$

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Studdy Solution

STEP 1

1. We have a system of linear equations with two variables $p$ and $q$ .
2. We are given two methods to eliminate a variable and solve the system.
3. We need to determine if both methods correctly solve the system.

STEP 2

1. Apply the first method: Multiply the first equation by 2 and subtract the second equation.
2. Apply the second method: Multiply the second equation by 2 and add it to the first equation.
3. Verify if both methods lead to a valid solution for the system.

STEP 3

Multiply the first equation $4p + 2q = 62$ by 2:
$2(4p + 2q) = 2 \times 62$
$8p + 4q = 124$

STEP 4

Subtract the second equation $8p - q = 59$ from the result:
$(8p + 4q) - (8p - q) = 124 - 59$
$8p + 4q - 8p + q = 65$
$5q = 65$

STEP 5

Solve for $q$ :
$q = \frac{65}{5} = 13$

STEP 6

Multiply the second equation $8p - q = 59$ by 2:
$2(8p - q) = 2 \times 59$
$16p - 2q = 118$

STEP 7

Add the result to the first equation $4p + 2q = 62$ :
$(16p - 2q) + (4p + 2q) = 118 + 62$
$16p - 2q + 4p + 2q = 180$
$20p = 180$

STEP 8

Solve for $p$ :
$p = \frac{180}{20} = 9$

STEP 9

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Studdy Solution

STEP 1

1. We have a system of linear equations with two variables p p p and q q q.2. We are given two methods to eliminate a variable and solve the system.3. We need to determine if both methods correctly solve the system.

STEP 2

1. Apply the first method: Multiply the first equation by 2 and subtract the second equation.2. Apply the second method: Multiply the second equation by 2 and add it to the first equation.3. Verify if both methods lead to a valid solution for the system.

STEP 3

Multiply the first equation 4p+2q=62 4p + 2q = 62 4p+2q=62 by 2:2(4p+2q)=2×62 2(4p + 2q) = 2 \times 62 2(4p+2q)=2×628p+4q=124 8p + 4q = 124 8p+4q=124

STEP 4

Subtract the second equation 8p−q=59 8p - q = 59 8p−q=59 from the result:(8p+4q)−(8p−q)=124−59 (8p + 4q) - (8p - q) = 124 - 59 (8p+4q)−(8p−q)=124−598p+4q−8p+q=65 8p + 4q - 8p + q = 65 8p+4q−8p+q=655q=65 5q = 65 5q=65

STEP 5

Solve for q q q:q=655=13 q = \frac{65}{5} = 13 q=565​=13

STEP 6

Multiply the second equation 8p−q=59 8p - q = 59 8p−q=59 by 2:2(8p−q)=2×59 2(8p - q) = 2 \times 59 2(8p−q)=2×5916p−2q=118 16p - 2q = 118 16p−2q=118

STEP 7

Add the result to the first equation 4p+2q=62 4p + 2q = 62 4p+2q=62:(16p−2q)+(4p+2q)=118+62 (16p - 2q) + (4p + 2q) = 118 + 62 (16p−2q)+(4p+2q)=118+6216p−2q+4p+2q=180 16p - 2q + 4p + 2q = 180 16p−2q+4p+2q=18020p=180 20p = 180 20p=180

STEP 8

Solve for p p p:p=18020=9 p = \frac{180}{20} = 9 p=20180​=9

STEP 9

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1. We have a system of linear equations with two variables $p$ and $q$ .
2. We are given two methods to eliminate a variable and solve the system.
3. We need to determine if both methods correctly solve the system.

1. Apply the first method: Multiply the first equation by 2 and subtract the second equation.
2. Apply the second method: Multiply the second equation by 2 and add it to the first equation.
3. Verify if both methods lead to a valid solution for the system.

Multiply the first equation $4p + 2q = 62$ by 2:
$2(4p + 2q) = 2 \times 62$
$8p + 4q = 124$

Subtract the second equation $8p - q = 59$ from the result:
$(8p + 4q) - (8p - q) = 124 - 59$
$8p + 4q - 8p + q = 65$
$5q = 65$

Solve for $q$ :
$q = \frac{65}{5} = 13$

Multiply the second equation $8p - q = 59$ by 2:
$2(8p - q) = 2 \times 59$
$16p - 2q = 118$

Add the result to the first equation $4p + 2q = 62$ :
$(16p - 2q) + (4p + 2q) = 118 + 62$
$16p - 2q + 4p + 2q = 180$
$20p = 180$

Solve for $p$ :
$p = \frac{180}{20} = 9$