I can help with a couple of them. You are allowed a maximum of 3 questions per request. You will get more responses if you split the 4-page worksheet into several requests. There are 15 questions so split them into 5 requests.
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1) Rewrite the equations in y = mx + b format (m = slope, b = y-intercept)
Independent: one solution ⇒ different slopes
Dependent: infinite solutions (same line) ⇒ same slope and y-intercept
Inconsistent: no solutions (parallel lines) ⇒ same slope but different y-intercepts
-21x - 3y = -24 → -3y = 21x - 24 → y = -7x + 8 ⇒ m = -7, b = 8
y = -7x + 7 ⇒ m = -7, b = 7
same slope but different y-intercepts = parallel lines = no solution
Answer: Inconsistent
12) Eliminate a variable for equations 1 and 2, then eliminate the same variable for equations 2 and 3. You will end up with two new equations and two variables. Solve that system by eliminating one of the variables (using any method) and solving for the remaining variable. Then plug that answer into one of the new equations. Then plug in both those answers into one of the original equations.
EQ1: x + 3y + 2z = 8 → 1(x + 3y + 2z = 8) → x + 3y + 2z = 8
EQ2: 3x + y + 3z = -10 → -3(3x + y + 3z = -10) →<u> -9x - 3y - 9z</u> = <u>30 </u>
NEW EQ 1,2: -8x -7z = 38
EQ2: 3x + y + 3z = -10 → 2(3x + y + 3z = -10) → 6x + 2y + 6z = -20
EQ3: -2x - 2y - z = 10 → 1(-2x - 2y - z = 10) → <u>-2x - 2y - z </u>= <u> 10 </u>
NEW EQ 2,3: 4x +5z = -10
NEW EQ 1,2: -8x - 7z = 38 → 1(-8x - 7z = 38) → -8x - 7z = 38
NEW EQ 2,3: 4x + 5z = -10 → 2(4x + 5z = -10) → <u> 8x + 10z </u>= <u>-20 </u>
3z = 18
z = 6
NEW EQ 2,3: 4x + 5z = -10 ⇒ 4x + 5(6) = -10 ⇒ 4x + 30 = -10 ⇒ 4x = -40 ⇒ x = -10
EQ1: x + 3y + 2z = 8 ⇒ (-10) + 3y + 2(6) = 8 ⇒ 3y + 2 = 8 ⇒ 3y = 6 ⇒ y = 2
Answer: x = -10, y = 2, z = 6
