Question

solving systems of equations by substitution

1. use substitution to determine the solution of

the system of equations.
$y = -2x - 7$
$2y - x = 1$
$\boldsymbol{(-1, 0)}$ $\boldsymbol{(-6, -\frac{5}{2})}$
$\boldsymbol{(\frac{21}{2}, -28)}$ $\boldsymbol{(-3, -1)}$

1. use substitution to determine the solution of the system

of equations. write the solution as an ordered pair.
$x + 2y = 14$
$y = 3x - 14$

1. use substitution to determine which system

is represented by the graph.
$\boldsymbol{2x + 4y = 28\\-2y = x + 28}$ $\boldsymbol{y = 3x + 10\\2x - 3y = -6}$
$\boldsymbol{y = 10x - 15\\-9x + 2y = 10}$ $\boldsymbol{2y = 2x + 5\\3x - 4y = -5}$

1. skyler buys 8 t-shirts and 5 hats for $220. the next

day, he buys 5 t-shirts and 1 hat for $112. how much
does each t-shirt and each hat cost? write a system
of equations that can be used to solve the problem.
then solve the problem.
system of equations: ____________
____________
t-shirt cost: __________ hat cost: __________

1. use substitution to determine whether the system

below has no solutions, infinitely many solutions,
or one solution.
$15x + 5y = 20$
$y = 8 - 3x$

Explanation:

Problem 1

Step1: Substitute $y=-2x-7$ into $2y-x=1$

$2(-2x-7) - x = 1$

Step2: Simplify and solve for $x$

$-4x -14 -x = 1 \implies -5x = 15 \implies x = -3$

Step3: Substitute $x=-3$ into $y=-2x-7$

$y = -2(-3) -7 = 6 -7 = -1$

Step1: Substitute $y=3x-14$ into $x+2y=14$

$x + 2(3x-14) = 14$

Step2: Simplify and solve for $x$

$x +6x -28 =14 \implies 7x=42 \implies x=6$

Step3: Substitute $x=6$ into $y=3x-14$

$y=3(6)-14=18-14=4$

Step1: Identify graph features: Two lines intersect at $(0,7)$

Test $(0,7)$ in each system:

Step2: Test Option A

$2(0)+4(7)=28$ (true); $-2(7)=0+28 \implies -14=28$ (false)

Step3: Test Option B

$7=10(0)-15 \implies7=-15$ (false)

Step4: Test Option C

$7=3(0)+10 \implies7=10$ (false)

Step5: Test Option D

$2(7)=2(0)+5 \implies14=5$ (false)
Correction: Recheck Option A first equation $2x+4y=28$ simplifies to $x+2y=14$; second equation $-2y=x+28$ simplifies to $x+2y=-28$. These are parallel (same slope, different intercepts), matching the graph's parallel lines.

Answer:

$\boldsymbol{D\ (-3, -1)}$

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Problem 2