Math / Algebra

Question4. $\left\{\begin{array}{l}-5 m+9 n=21 \\ 2 m+2 n=14\end{array}\right.$

Studdy Solution

STEP 1

1. The given problem involves solving a system of linear equations with two variables, $m$ and $n$ .
2. The system of equations is: $ \begin{cases} -5m + 9n = 21 \\ 2m + 2n = 14 \end{cases} \]
3. The solution will be found by using methods such as substitution or elimination.

STEP 2

1. Simplify the equations if necessary.
2. Use the elimination method to solve for one of the variables.
3. Substitute the found variable back into one of the original equations to find the other variable.
4. Verify the solution by plugging both variables back into the original equations.

STEP 3

Simplify the second equation by dividing by 2:
$2m + 2n = 14 \implies m + n = 7$

STEP 4

Use the elimination method. Multiply the simplified second equation by 5 to facilitate elimination:
$5(m + n) = 5 \cdot 7 \implies 5m + 5n = 35$

STEP 5

Add the modified second equation to the first equation to eliminate $m$ :
$-5m + 9n + 5m + 5n = 21 + 35$
$14n = 56$

STEP 6

Solve for $n$ by dividing by 14:
$n = \frac{56}{14} = 4$

STEP 7

Substitute $n = 4$ back into the simplified second equation to find $m$ :
$m + 4 = 7 \implies m = 7 - 4 = 3$

STEP 8

Verify the solution by substituting $m = 3$ and $n = 4$ into the original equations:
1. Check the first equation: $ -5m + 9n = 21 \implies -5(3) + 9(4) = -15 + 36 = 21 \quad \text{(True)} \]
2. Check the second equation: $ 2m + 2n = 14 \implies 2(3) + 2(4) = 6 + 8 = 14 \quad \text{(True)} \]
The solution to the system of equations is:
$m = 3, \quad n = 4$

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Question4. $\left\{\begin{array}{l}-5 m+9 n=21 \\ 2 m+2 n=14\end{array}\right.$

Studdy Solution

STEP 1

1. The given problem involves solving a system of linear equations with two variables, $m$ and $n$ .
2. The system of equations is: $ \begin{cases} -5m + 9n = 21 \\ 2m + 2n = 14 \end{cases} \]
3. The solution will be found by using methods such as substitution or elimination.

STEP 2

1. Simplify the equations if necessary.
2. Use the elimination method to solve for one of the variables.
3. Substitute the found variable back into one of the original equations to find the other variable.
4. Verify the solution by plugging both variables back into the original equations.

STEP 3

Simplify the second equation by dividing by 2:
$2m + 2n = 14 \implies m + n = 7$

STEP 4

Use the elimination method. Multiply the simplified second equation by 5 to facilitate elimination:
$5(m + n) = 5 \cdot 7 \implies 5m + 5n = 35$

STEP 5

Add the modified second equation to the first equation to eliminate $m$ :
$-5m + 9n + 5m + 5n = 21 + 35$
$14n = 56$

STEP 6

Solve for $n$ by dividing by 14:
$n = \frac{56}{14} = 4$

STEP 7

Substitute $n = 4$ back into the simplified second equation to find $m$ :
$m + 4 = 7 \implies m = 7 - 4 = 3$

STEP 8

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Question4. {−5m+9n=212m+2n=14\left\{\begin{array}{l}-5 m+9 n=21 \\ 2 m+2 n=14\end{array}\right.{−5m+9n=212m+2n=14​

Studdy Solution

STEP 1

1. The given problem involves solving a system of linear equations with two variables, mmm and nnn.2. The system of equations is: $ \begin{cases} -5m + 9n = 21 \\ 2m + 2n = 14 \end{cases} \]3. The solution will be found by using methods such as substitution or elimination.

STEP 2

1. Simplify the equations if necessary.2. Use the elimination method to solve for one of the variables.3. Substitute the found variable back into one of the original equations to find the other variable.4. Verify the solution by plugging both variables back into the original equations.

STEP 3

Simplify the second equation by dividing by 2:2m+2n=14 ⟹ m+n=72m + 2n = 14 \implies m + n = 72m+2n=14⟹m+n=7

STEP 4

Use the elimination method. Multiply the simplified second equation by 5 to facilitate elimination:5(m+n)=5⋅7 ⟹ 5m+5n=355(m + n) = 5 \cdot 7 \implies 5m + 5n = 355(m+n)=5⋅7⟹5m+5n=35

STEP 5

Add the modified second equation to the first equation to eliminate mmm:−5m+9n+5m+5n=21+35-5m + 9n + 5m + 5n = 21 + 35−5m+9n+5m+5n=21+3514n=5614n = 5614n=56

STEP 6

Solve for nnn by dividing by 14:n=5614=4n = \frac{56}{14} = 4n=1456​=4

STEP 7

Substitute n=4n = 4n=4 back into the simplified second equation to find mmm:m+4=7 ⟹ m=7−4=3m + 4 = 7 \implies m = 7 - 4 = 3m+4=7⟹m=7−4=3

STEP 8

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Question4. $\left\{\begin{array}{l}-5 m+9 n=21 \\ 2 m+2 n=14\end{array}\right.$

1. The given problem involves solving a system of linear equations with two variables, $m$ and $n$ .
2. The system of equations is: $ \begin{cases} -5m + 9n = 21 \\ 2m + 2n = 14 \end{cases} \]
3. The solution will be found by using methods such as substitution or elimination.

1. Simplify the equations if necessary.
2. Use the elimination method to solve for one of the variables.
3. Substitute the found variable back into one of the original equations to find the other variable.
4. Verify the solution by plugging both variables back into the original equations.

Simplify the second equation by dividing by 2:
$2m + 2n = 14 \implies m + n = 7$

Use the elimination method. Multiply the simplified second equation by 5 to facilitate elimination:
$5(m + n) = 5 \cdot 7 \implies 5m + 5n = 35$

Add the modified second equation to the first equation to eliminate $m$ :
$-5m + 9n + 5m + 5n = 21 + 35$
$14n = 56$

Solve for $n$ by dividing by 14:
$n = \frac{56}{14} = 4$

Substitute $n = 4$ back into the simplified second equation to find $m$ :
$m + 4 = 7 \implies m = 7 - 4 = 3$