Question

part 1 - a bunch of systems
solve each system of equations without graphing and show your reasoning. then, check your solutions.

a. $2x + 3y = 7$
$-2x + 4y = 14$

b. $2x + 3y = 7$
$3x - 3y = 3$

c. $2x + 3y = 5$
$2x + 4y = 9$

d. $2x + 3y = 16$
$6x - 5y = 20$

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Explanation:

a. System 1

Step1: Add equations to eliminate $x$

$$\begin{align*} (2x + 3y) + (-2x + 4y) &= 7 + 14 \\ 7y &= 21 \end{align*}$$

Step2: Solve for $y$

$y = \frac{21}{7} = 3$

Step3: Substitute $y=3$ into first equation

$2x + 3(3) = 7$
$2x + 9 = 7$

Step4: Solve for $x$

$2x = 7 - 9 = -2$
$x = \frac{-2}{2} = -1$

Step5: Verify solution

Substitute $x=-1, y=3$ into second equation:
$-2(-1) + 4(3) = 2 + 12 = 14$, which matches.

b. System 2

Step1: Add equations to eliminate $y$

$$\begin{align*} (2x + 3y) + (3x - 3y) &= 7 + 3 \\ 5x &= 10 \end{align*}$$

Step2: Solve for $x$

$x = \frac{10}{5} = 2$

Step3: Substitute $x=2$ into first equation

$2(2) + 3y = 7$
$4 + 3y = 7$

Step4: Solve for $y$

$3y = 7 - 4 = 3$
$y = \frac{3}{3} = 1$

Step5: Verify solution

Substitute $x=2, y=1$ into second equation:
$3(2) - 3(1) = 6 - 3 = 3$, which matches.

c. System 3

Step1: Subtract first equation from second to eliminate $x$

$$\begin{align*} (2x + 4y) - (2x + 3y) &= 9 - 5 \\ y &= 4 \end{align*}$$

Step2: Substitute $y=4$ into first equation

$2x + 3(4) = 5$
$2x + 12 = 5$

Step3: Solve for $x$

$2x = 5 - 12 = -7$
$x = \frac{-7}{2} = -3.5$

Step4: Verify solution

Substitute $x=-3.5, y=4$ into second equation:
$2(-3.5) + 4(4) = -7 + 16 = 9$, which matches.

d. System 4

Step1: Multiply first equation by 3 to align $x$ terms

$3(2x + 3y) = 3(16)$
$6x + 9y = 48$

Step2: Subtract second equation from new equation

$$\begin{align*} (6x + 9y) - (6x - 5y) &= 48 - 20 \\ 14y &= 28 \end{align*}$$

Step3: Solve for $y$

$y = \frac{28}{14} = 2$

Step4: Substitute $y=2$ into first equation

$2x + 3(2) = 16$
$2x + 6 = 16$

Step5: Solve for $x$

$2x = 16 - 6 = 10$
$x = \frac{10}{2} = 5$

Step6: Verify solution

Substitute $x=5, y=2$ into second equation:
$6(5) - 5(2) = 30 - 10 = 20$, which matches.

Answer:

a. $x=-1$, $y=3$
b. $x=2$, $y=1$
c. $x=-3.5$, $y=4$
d. $x=5$, $y=2$