1) The answer is: [B]: r = 5 .
Explanation:
Given: 7r − 7 = 2r + 18 ; Round your answer to the nearest tenth, if necessary.
Since "r" is the only variable given, let us assume we want to solve for "r" (instead of "x").
→ Subtract "2r" from EACH SIDE of the equation; and & add "7" to EACH SIDE of the equation:
→ 7r − 7 − 2r + 7 = 2r + 18 − 2r + 7 ; to get: → 5r = 25 ;
→ Now, divide EACH SIDE of the equation by "5"; to isolate "r" on one side of the equation; and to solve for "r" :
→ 5r / 5 = 25 / 5 → r = 5 → which is: "Answer choice: [B]".
Let us check our answer, by plugging in "5" for "r" in the original equation:
→ 7r − 7 = 2r + 18 ; → 7(5) − 7 =? 2(5) + 18? ;
→ 35 − 7 =? 10 + 18 ?; → 28 =? 28? Yes!
2) The answer is: [D]: x = 2 .
Explanation:
Given: 2x + 12 = 18 − x ; Solve for "x" (round to nearest tenth, if necessary).
→ Add "x" to EACH SIDE of the equation, & subtract "12" from EACH SIDE of the equation: → 2x + 12 + x − 12 = 18 − x + x − 12 ;
→ To get: 3x = 6 ; → Divide EACH SIDE of the equation by "3";
to isolate "x" on one side of the equation; and to solve for "x":
→ 3x / 3 = 6 / 3 ; → x = 2 ; which is: "Answer choice: [D]".
Let us check our answer, by plugging in "2" for "x" in the original equation:
→ 2x + 12 = 18 − x ; → 2(2) + 12 =? 18 − 2 ?
→ 4 + 12 =? 18 − 2 ? ; → 16 =? 16? Yes!
3) The answer is: [A]: x = -3 .
Explanation:
Given: 8x − 3 = 15x + 18 ; Solve for "x". Round your answer to the nearest tenth, if necessary.
→ Subtract "8x" from EACH SIDE of the equation, & add "3" to EACH SIDE of the equation:
→ 8x − 3 − 8x + 3 = 15x + 18 − 8x + 3 ; to get:
→ 0 = 7x + 21 ; → Subtract "21" from EACH SIDE of the equation;
→ 0 − 21 = 7x + 21 − 21 ; to get:
→ -21 = 7x ; Now divide EACH SIDE of the equation by "7";
to isolate "x" on one side of the equation; & to solve for "x":
→ = -21 / 7 = 7x / 7 ; → -3 = x ; which is "Answer choice: [A]."
Let us check our answer, by plugging in "-3" for "x" in the original equation:
→ 8x − 3 = 15x + 18 ; → 8(-3) − 3 =? 15(-3) + 18 ?;
→ -24 − 3 =? -45 + 18 ? ; → -27 =? -27? Yes!
4) The answer is: [C]: y = 11 .
Explanation:
Given: 6y − 6 = 4y + 16 ; Solve for "y"; Round to the nearest tenth, if necessary.
(Note: Since "y" is the only variable given; assume we are to solve for "y" instead of "x").
→ Subtract "4y" from EACH SIDE of the equation, & add "6" to EACH SIDE of the equation; → 6y − 6 − 4y + 6 = 4y + 16 − 4y + 6 ; to get:
→ 2y = 22 ; Now, divide EACH SIDE of the equation by "2"; to isolate "y" one side of the equation; and to solve for "y" ;
→ 2y / 2 = 22 / 2 ; → y = 11 → which is "Answer choice: [C]".
Let us check our answer, by plugging in "11" for "y" in the original equation:
→ 6y − 6 = 4y + 16 ; → 6(11) − 6 =? 4(11) + 16 ?
→ 66 − 6 =? 44 + 16 ? → 60 =? 60 ? Yes!
5) The answer is: [B]: x = -11 .
Explanation:
Given: 3(x − 4) = 5(x + 2) ; Solve for "x". Round to the nearest tenth, if necessary.
→Note the "distributive property of multiplication":
a*(b + c) = ab + ac ; and: a*(b − c) = ab − ac ;
→ Let us expand EACH SIDE of our given equation.
→Start with the "left-hand side":
3(x − 4) = (3*x) − (3*4) = 3x − 12;
→Now let us expand the "right-hand side" of the given equation:
→ 5(x + 2) = (5*x) + (5*2) = 5x + 10 ;
→Now, we can rewrite the original equation:
→ 3(x − 4) = 5(x + 2) ; by substituting the expanded values for EACH SIDE of the question: → 3x − 12 = 5x + 10 ;
→ Subtract "3x" from EACH SIDE of the equation; and add "12" to EACH SIDE of the equation: → 3x − 12 − 3x + 12 = 5x + 10 − 3x + 12 ; to get:
→ 0 = 2x + 22; → Now subtract "22" from EACH SIDE of the equation:
→ 0 − 22 = 2x + 22 − 22 ; to get: → -22 = 2x ;
→ Divide EACH SIDE of the equation by "2"; to isolate "x" on one side of the equation; & to solve for "x" ;
→ -22 / 2 = 2x /2 ; → -11 = x ; which is "Answer choice: [B]".
Let us check our answer, by plugging in "-11" for "x" in the original equation:
→ 3(x − 4) = 5(x + 2) ; → 3(-11 − 4) =? 5(-11 + 2) ? ;
→3(-15) =? 5(-9) ? ; → -45 =? -45 ? Yes!
Hope these answers and explanations are helpful. Best of luck!
