Math / Algebra

QuestionSolve by the substitution method. $\begin{array}{l} 2 x+3 y=21 \\ x=15-2 y \end{array}$
Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The solution set of the system is $\square$ \}. (Simplify your answer. Type an ordered pair.) B. There are infinitely many solutions. C. The solution set is the empty set.

Studdy Solution

STEP 1

1. The system of equations involves two variables, $x$ and $y$ .
2. We are to solve the system using the substitution method.
3. We assume the equations are consistent and have a unique solution unless proven otherwise.

STEP 2

1. Substitute the expression for $x$ from the second equation into the first equation.
2. Solve the resulting single-variable equation for $y$ .
3. Substitute the value of $y$ back into the expression for $x$ to find $x$ .
4. Verify the solution by plugging the values of $x$ and $y$ back into the original equations.

STEP 3

Substitute the expression $x = 15 - 2y$ from the second equation into the first equation.
$2x + 3y = 21$ Substitute $x = 15 - 2y$ :
$2(15 - 2y) + 3y = 21$

STEP 4

Simplify and solve the resulting equation for $y$ .
$2(15 - 2y) + 3y = 21$
Distribute the $2$ :
$30 - 4y + 3y = 21$
Combine like terms:
$30 - y = 21$
Solve for $y$ :
$-y = 21 - 30$
$-y = -9$
$y = 9$

STEP 5

Substitute $y = 9$ back into the expression $x = 15 - 2y$ to find $x$ .
$x = 15 - 2(9)$
Simplify:
$x = 15 - 18$
$x = -3$

STEP 6

Verify the solution $(x, y) = (-3, 9)$ by substituting back into the original equations.
First equation:
$2x + 3y = 21$
$2(-3) + 3(9) = 21$
$-6 + 27 = 21$
$21 = 21$ (True)
Second equation:
$x = 15 - 2y$
$-3 = 15 - 2(9)$
$-3 = 15 - 18$
$-3 = -3$ (True)
Both equations are satisfied, thus the solution is correct.
The solution set of the system is $\{(-3, 9)\}$ .

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Studdy Solution

STEP 1

1. The system of equations involves two variables, $x$ and $y$ .
2. We are to solve the system using the substitution method.
3. We assume the equations are consistent and have a unique solution unless proven otherwise.

STEP 2

STEP 3

Substitute the expression $x = 15 - 2y$ from the second equation into the first equation.
$2x + 3y = 21$ Substitute $x = 15 - 2y$ :
$2(15 - 2y) + 3y = 21$

STEP 4

Simplify and solve the resulting equation for $y$ .
$2(15 - 2y) + 3y = 21$
Distribute the $2$ :
$30 - 4y + 3y = 21$
Combine like terms:
$30 - y = 21$
Solve for $y$ :
$-y = 21 - 30$
$-y = -9$
$y = 9$

STEP 5

Substitute $y = 9$ back into the expression $x = 15 - 2y$ to find $x$ .
$x = 15 - 2(9)$
Simplify:
$x = 15 - 18$
$x = -3$

STEP 6

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Studdy Solution

STEP 1

1. The system of equations involves two variables, xxx and yyy.2. We are to solve the system using the substitution method.3. We assume the equations are consistent and have a unique solution unless proven otherwise.

STEP 2

STEP 3

Substitute the expression x=15−2yx = 15 - 2yx=15−2y from the second equation into the first equation.2x+3y=212x + 3y = 212x+3y=21 Substitute x=15−2yx = 15 - 2yx=15−2y:2(15−2y)+3y=212(15 - 2y) + 3y = 212(15−2y)+3y=21

STEP 4

Simplify and solve the resulting equation for yyy.2(15−2y)+3y=212(15 - 2y) + 3y = 212(15−2y)+3y=21Distribute the 222:30−4y+3y=2130 - 4y + 3y = 2130−4y+3y=21Combine like terms:30−y=2130 - y = 2130−y=21Solve for yyy:−y=21−30-y = 21 - 30−y=21−30−y=−9-y = -9−y=−9y=9y = 9y=9

STEP 5

Substitute y=9y = 9y=9 back into the expression x=15−2yx = 15 - 2yx=15−2y to find xxx.x=15−2(9)x = 15 - 2(9)x=15−2(9)Simplify:x=15−18x = 15 - 18x=15−18x=−3x = -3x=−3

STEP 6

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1. The system of equations involves two variables, $x$ and $y$ .
2. We are to solve the system using the substitution method.
3. We assume the equations are consistent and have a unique solution unless proven otherwise.

Substitute the expression $x = 15 - 2y$ from the second equation into the first equation.
$2x + 3y = 21$ Substitute $x = 15 - 2y$ :
$2(15 - 2y) + 3y = 21$

Simplify and solve the resulting equation for $y$ .
$2(15 - 2y) + 3y = 21$
Distribute the $2$ :
$30 - 4y + 3y = 21$
Combine like terms:
$30 - y = 21$
Solve for $y$ :
$-y = 21 - 30$
$-y = -9$
$y = 9$

Substitute $y = 9$ back into the expression $x = 15 - 2y$ to find $x$ .
$x = 15 - 2(9)$
Simplify:
$x = 15 - 18$
$x = -3$