8th Grade Math Solving Linear Equations Online Practice
1. The distance formula can be written as d=rt
, where d= distance traveled, r= the rate of speed, and t= the length of time traveled.
Johan travels 35 miles by bicycle in 1.75 hours. What is Johan’s rate of speed in miles per hour? Explain the solution to the real-world problem.
2. Raji draws a square with a side length of x+5cm. Ava draws a regular hexagon with a side length of x−1cm. The two figures have the same perimeter. Write an equation that allows you to solve for x. What is the value of x?
3. Juwan is thinking of a number. Twice the sum of Juwan's number and 12 is equal to four less than three times Juwan's number. What number is Juwan thinking of?
4. Which of the following equations has exactly one solution?
A. -5z+4=5z+4
B. -5z+3=-5z+3
C. 5z+4=5z+4
D. 5z+4=5z+3
5. Solve for x to determine if the equation has one, none, or infinite solutions.
3(2x+2)−4=−2x+10
6. Which answer gives all the values of z for which the inequality 3x+2≥9−4x is true?
A. x≥1
x greater-than-or-equal-to 1
B. x=1
x equals 1
C. x≤1
x less-than-or-equal-to 1
D. x>1
x greater-than-or-equal-to 1
7. Maluwa solves the inequality 3x+9≥6. Which statement explains the steps she uses to find the solution set of the inequality?
A. Maluwa subtracts 9 from each side of the inequality, divides both sides by 3, and reverses the inequality symbol to obtain the solution ≤−1.
B. Maluwa subtracts 9 from each side of the inequality and then divides both sides by 3 to obtain the solution x≥−1.
C. Maluwa subtracts 9 from both sides of the inequality and then multiplies both sides by 3 to obtain the solution x≥−9.
D. Maluwa adds 9 to each side of the inequality and then divides both sides by 3 to obtain the solution x≥5.
8. Which of the following is the correct solution to the inequality Cx+4≤18, where C is a number greater than 0?
A. C ≤ 14/x
B. x ≤ 14
C. x ≤ 14/C
D. Cx ≤ 14
1. To find Johan's rate of speed, we can use the distance formula d=rt. Given that d = 35 miles and t = 1.75 hours, we can substitute these values into the formula to get 35 = r * 1.75. To solve for r, we divide both sides of the equation by 1.75, resulting in r = 20 miles per hour. Therefore, Johan's rate of speed is 20 miles per hour.
2. The perimeter of a square with side length x+5cm is equal to the perimeter of a regular hexagon with side length x-1cm. The perimeter of a square is 4 * side length, while the perimeter of a regular hexagon is 6 * side length. Therefore, we can write the equation 4(x+5) = 6(x-1) to solve for x. Expanding and simplifying this equation gives us 4x + 20 = 6x - 6. By subtracting 4x and adding 6 to both sides, we get 26 = 2x. Dividing by 2, we find x = 13. Thus, the value of x is 13.
3. Let's represent the number Juwan is thinking of as n. The problem states that twice the sum of Juwan's number and 12 is equal to four less than three times Juwan's number. This can be written as 2(n + 12) = 3n - 4. Expanding and simplifying, we have 2n + 24 = 3n - 4. By subtracting 2n from both sides and adding 4, we get 28 = n. Therefore, Juwan is thinking of the number 28.
4. The equation that has exactly one solution is D. 5z + 4 = 5z + 3. This equation has one solution because when we subtract 5z from both sides, we get 4 = 3, which is not true. Therefore, there are no values of z that satisfy this equation, resulting in exactly one solution.
5. To solve for x, we can begin by distributing the 3 on the left side of the equation. This gives us 6x + 6 - 4 = -2x + 10. Simplifying further, we have 6x + 2 = -2x + 10. By adding 2x to both sides and subtracting 2 from both sides, we obtain 8x = 8. Dividing both sides by 8, we find x = 1. Therefore, the equation has one solution, which is x = 1.
6. The statement that gives all the values of z for which the inequality 3x + 2 ≥ 9 - 4x is true is C. x ≤ 1. To solve this inequality, we can first combine like terms and bring the variables to one side. This gives us 7x + 2 ≥ 9. By subtracting 2 from both sides, we get 7x ≥ 7. Finally, by dividing both sides by 7, we find x ≤ 1. Therefore, the inequality is true when x is less than or equal to 1.
7. Maluwa solves the inequality 3x + 9 ≥ 6 by subtracting 9 from each side of the inequality, resulting in 3x ≥ -3. Then, she divides both sides by 3 to get x ≥ -1. Therefore, the correct statement is B. Maluwa subtracts 9 from each side of the inequality and then divides both sides by 3 to obtain the solution x ≥ -1.
8. The correct solution to the inequality Cx + 4 ≤ 18, where C is a number greater than 0, is C. x ≤ 14/C. To solve this inequality, we can start by subtracting 4 from both sides, giving us Cx ≤ 14. Finally, by dividing both sides by C, we find x ≤ 14/C. Therefore, the correct solution is x ≤ 14/C.
4. Is -5x + 4= 5x +4
6. A. x greater than equal to 1
4. The equation -5x + 4 = 5x + 4 has infinite solutions. This is because when we subtract 5x from both sides and subtract 4 from both sides, we are left with 0 = 0, which is always true. Therefore, any value of x will satisfy this equation.
6. The correct answer is A. x ≥ 1. The inequality 3x + 2 ≥ 9 - 4x can be solved by combining like terms and bringing all the variables to one side. This gives us 7x + 2 ≥ 9. By subtracting 2 from both sides, we get 7x ≥ 7. Finally, by dividing both sides by 7, we find x ≥ 1. Therefore, the inequality is true when x is greater than or equal to 1.
1. To find Johan's rate of speed, we can use the distance formula, which is d = rt. Given that Johan travels 35 miles in 1.75 hours, we can substitute these values into the formula:
35 = r * 1.75
To find the rate of speed, we need to solve for r. Divide both sides of the equation by 1.75:
35/1.75 = r
Simplify the left side:
20 = r
Therefore, Johan's rate of speed is 20 miles per hour.
2. The perimeter of a square is given by the formula P = 4s, where P is the perimeter and s is the side length. Similarly, the perimeter of a regular hexagon is given by the formula P = 6s, where P is the perimeter and s is the side length.
Since the two figures have the same perimeter, we can set up the equation:
4(x+5) = 6(x-1)
Simplify both sides of the equation:
4x + 20 = 6x - 6
Combine like terms:
20 + 6 = 6x - 4x - 6
Simplify further:
26 = 2x - 6
Add 6 to both sides of the equation:
26 + 6 = 2x - 6 + 6
Simplify:
32 = 2x
Divide both sides of the equation by 2:
32/2 = 2x/2
Simplify:
16 = x
Therefore, the value of x is 16.
3. Let's set up an equation to represent the problem. Let's call the number Juwan is thinking of "n".
Twice the sum of Juwan's number and 12 is equal to four less than three times Juwan's number. Mathematically, we can express this as:
2(n+12) = 3n - 4
Now, we can solve for n. Distribute the 2 on the left side:
2n + 24 = 3n - 4
Subtract 2n from both sides to isolate the variable:
24 = n - 4
Add 4 to both sides:
24 + 4 = n - 4 + 4
Simplify:
28 = n
Therefore, Juwan is thinking of the number 28.
4. To determine which equation has exactly one solution, we need to look for the equation where the variables (z in this case) cancel out or balance on both sides of the equation.
The equation -5z+4=5z+4 has variables (z) on both sides of the equation but they balance out. If we subtract 5z from both sides, we get:
-5z + 4 - 5z = 5z + 4 - 5z
-10z + 4 = 4
Simplify:
-10z = 0
Divide both sides by -10:
z = 0
Therefore, the equation -5z+4=5z+4 has exactly one solution, z = 0.
5. To solve for x in the equation 3(2x+2)−4 = −2x+10, we will simplify the equation step by step and then solve for x.
First, distribute the 3 to the terms inside the parentheses:
6x + 6 - 4 = -2x + 10
Combine like terms on the left side:
6x + 2 = -2x + 10
Add 2x to both sides to isolate the variable:
6x + 2x + 2 = 10
Combine like terms on the right side:
8x + 2 = 10
Subtract 2 from both sides:
8x = 8
Divide both sides by 8:
x = 1
Therefore, the equation has one solution, x = 1.
6. To find the values of z that satisfy the inequality 3x+2 ≥ 9-4x, we will solve the inequality step by step.
First, combine like terms on the left side:
7x + 2 ≥ 9
Subtract 2 from both sides to isolate the variable:
7x ≥ 7
Divide both sides by 7 to solve for x:
x ≥ 1
Therefore, the correct answer is A. x ≥ 1, which means all values of x greater than or equal to 1 satisfy the inequality.
7. To find the solution set of the inequality 3x + 9 ≥ 6, we will solve the inequality step by step.
First, subtract 9 from both sides to isolate the variable:
3x ≥ 6 - 9
Simplify:
3x ≥ -3
Divide both sides by 3 to solve for x:
x ≥ -1
Therefore, the correct answer is B. Maluwa subtracts 9 from each side of the inequality and then divides both sides by 3 to obtain the solution x ≥ -1.
8. To solve the inequality Cx + 4 ≤ 18, where C is a number greater than 0, we will subtract 4 from both sides to isolate the variable:
Cx ≤ 18 - 4
Simplify:
Cx ≤ 14
Divide both sides by C to solve for x.
Since we know that C is a number greater than 0, we do not need to change the direction of the inequality sign:
x ≤ 14/C
Therefore, the correct answer is C. x ≤ 14/C, where C is a number greater than 0.