Question

solve each system using elimination.

1. $2x + 5y = 2$

$3x - 5y = 53$

1. $4x + 2y = 34$

$10x - 4y = -5$

1. $11x - 13y = 09$

$-11x + 13y = 107$

1. $3x + 6y = 42$

$-7x + 8y = -109$

Explanation:

Problem 9

Step1: Add the two equations

$2x + 5y + 3x - 5y = 2 + 53$

Step2: Simplify to solve for x

$5x = 55 \implies x = \frac{55}{5} = 11$

Step3: Substitute x=11 into first equation

$2(11) + 5y = 2$

Step4: Solve for y

$22 + 5y = 2 \implies 5y = 2 - 22 = -20 \implies y = \frac{-20}{5} = -4$

---

Problem 10

Step1: Multiply first equation by 2

$2(4x + 2y) = 2(34) \implies 8x + 4y = 68$

Step2: Add to second equation

$8x + 4y + 10x - 4y = 68 + (-5)$

Step3: Simplify to solve for x

$18x = 63 \implies x = \frac{63}{18} = \frac{7}{2} = 3.5$

Step4: Substitute x=3.5 into first equation

$4(3.5) + 2y = 34$

Step5: Solve for y

$14 + 2y = 34 \implies 2y = 34 - 14 = 20 \implies y = 10$

---

Problem 11 (corrected first equation: $11x - 13y = 0$)

Step1: Add the two equations

$11x - 13y + (-11x + 13y) = 0 + 107$

Step2: Simplify and analyze

$0 = 107$ (This is a contradiction, so no solution exists.)

---

Problem 12

Step1: Simplify first equation by dividing by 3

$x + 2y = 14 \implies x = 14 - 2y$

Step2: Substitute x into second equation

$-7(14 - 2y) + 8y = -109$

Step3: Expand and simplify

$-98 + 14y + 8y = -109 \implies 22y = -109 + 98 = -11$

Step4: Solve for y

$y = \frac{-11}{22} = -\frac{1}{2} = -0.5$

Step5: Substitute y=-0.5 into simplified first equation

$x = 14 - 2(-0.5) = 14 + 1 = 15$

Answer:

1. $x=11$, $y=-4$
2. $x=\frac{7}{2}$, $y=10$
3. No solution (contradiction)
4. $x=15$, $y=-\frac{1}{2}$