Question

what is the value of x of the systems of equations given below?

1. (-x - 4y = 4) (-2x + 9y = -26)
2. (-5x - 4y = -27) (-10x - 2y = -6)

what is the solution of the system of equations?

1. (y = -5x - 22) (3x + 3y = -18)
2. (6x - 2y = -4) (y = 3x + 2)

Explanation:

Problem 22:

Step1: Isolate $x$ from first eq.

$x = -4y - 4$

Step2: Substitute $x$ into second eq.

$-2(-4y - 4) + 9y = -26$

Step3: Simplify to solve for $y$

$8y + 8 + 9y = -26 \implies 17y = -34 \implies y = -2$

Step4: Substitute $y$ back for $x$

$x = -4(-2) - 4 = 8 - 4 = 4$

Problem 23:

Step1: Multiply first eq. by 2

$-10x - 8y = -54$

Step2: Subtract second eq. from new eq.

$(-10x - 8y) - (-10x - 2y) = -54 - (-6) \implies -6y = -48 \implies y = 8$

Step3: Substitute $y$ into first eq.

$-5x - 4(8) = -27 \implies -5x - 32 = -27 \implies -5x = 5 \implies x = -1$

Problem 24:

Step1: Substitute $y$ into second eq.

$3x + 3(-5x - 22) = -18$

Step2: Simplify to solve for $x$

$3x - 15x - 66 = -18 \implies -12x = 48 \implies x = -4$

Step3: Substitute $x$ back for $y$

$y = -5(-4) - 22 = 20 - 22 = -2$

Problem 25:

Step1: Substitute $y$ into first eq.

$6x - 2(3x + 2) = -4$

Step2: Simplify and check consistency

$6x - 6x - 4 = -4 \implies -4 = -4$

Answer:

1. $x = 4$
2. $x = -1$
3. $x = -4$, $y = -2$
4. Infinitely many solutions (all pairs where $y=3x+2$)