💡 9th Grade Other: Decuxyeurguf Practice Questions

To solve for \(x\), we need to isolate \(x\) on one side of the equation.

• Step 1: Identify the operation.
  The number 15 is being subtracted from \(x\).
• Step 2: Perform the inverse operation.
  To undo subtraction, we add. Add 15 to both sides of the equation.
  \[ x - 15 + 15 = 23 + 15 \]
• Step 3: Simplify.
  \[ x = 38 \]

✅ Solution: The value of \(x\) is 38.

Let's solve this two-step linear equation for \(y\).

• Step 1: Isolate the term with the variable.
  Subtract 7 from both sides of the equation to move the constant term.
  \[ 3y + 7 - 7 = 22 - 7 \]
  \[ 3y = 15 \]
• Step 2: Isolate the variable.
  The variable \(y\) is being multiplied by 3. To undo multiplication, we divide. Divide both sides by 3.
  \[ \frac{3y}{3} = \frac{15}{3} \]
• Step 3: Simplify.
  \[ y = 5 \]

✅ Solution: The value of \(y\) is 5.

To solve this equation, we need to gather all terms with \(m\) on one side and all constant terms on the other.

• Step 1: Move variable terms to one side.
  Subtract \(2m\) from both sides of the equation.
  \[ 5m - 2m - 8 = 2m - 2m + 10 \]
  \[ 3m - 8 = 10 \]
• Step 2: Move constant terms to the other side.
  Add 8 to both sides of the equation.
  \[ 3m - 8 + 8 = 10 + 8 \]
  \[ 3m = 18 \]
• Step 3: Isolate the variable.
  Divide both sides by 3.
  \[ \frac{3m}{3} = \frac{18}{3} \]
• Step 4: Simplify.
  \[ m = 6 \]

✅ Solution: The value of \(m\) is 6.

Solving inequalities is similar to solving equations, with one crucial difference!

• Step 1: Isolate the term with the variable.
  Subtract 5 from both sides of the inequality.
  \[ -2k + 5 - 5 \leq 11 - 5 \]
  \[ -2k \leq 6 \]
• Step 2: Isolate the variable.
  Divide both sides by -2. 📌 Important: When you multiply or divide an inequality by a negative number, you must reverse the inequality sign.
  \[ \frac{-2k}{-2} \geq \frac{6}{-2} \]
• Step 3: Simplify.
  \[ k \geq -3 \]
• Step 4: Graph the solution.
  On a number line, place a closed circle at -3 (because \(k\) can be equal to -3). Draw an arrow extending to the right, indicating all numbers greater than or equal to -3.
  

  (Imagine a number line with a closed circle at -3 and an arrow pointing to the right.)

✅ Solution: \(k \geq -3\).

Let's analyze the student's steps to find any errors.

• Error Identification:
  The error occurs in Step 1. The student incorrectly applied the distributive property. The 4 outside the parenthesis should be multiplied by BOTH terms inside the parenthesis, not just the first.
  The correct application of the distributive property is \(4(x - 3) = 4 \cdot x - 4 \cdot 3 = 4x - 12\).
• Correct Solution:
  Step 1: Apply the Distributive Property.
  \[ 4(x - 3) = 2x + 6 \]
  \[ 4x - 12 = 2x + 6 \]
• Step 2: Move variable terms to one side.
  Subtract \(2x\) from both sides.
  \[ 4x - 2x - 12 = 2x - 2x + 6 \]
  \[ 2x - 12 = 6 \]
• Step 3: Move constant terms to the other side.
  Add 12 to both sides.
  \[ 2x - 12 + 12 = 6 + 12 \]
  \[ 2x = 18 \]
• Step 4: Isolate the variable.
  Divide both sides by 2.
  \[ \frac{2x}{2} = \frac{18}{2} \]
• Step 5: Simplify.
  \[ x = 9 \]

✅ Solution: The error was in applying the distributive property. The correct solution is \(x = 9\).

Let's set up a linear equation to represent Sarah's savings plan.

• Step 1: Define the variable.
  Let \(w\) represent the number of weeks Sarah saves.
• Step 2: Formulate the equation.
  Sarah's total savings will be her initial savings plus the amount she saves each week.
  Initial savings: 70
  Savings per week: 15
  Total cost of bicycle: 250
  So, the equation is:
  \[ 70 + 15w = 250 \]
• Step 3: Solve the equation.
  First, subtract 70 from both sides:
  \[ 15w = 250 - 70 \]
  \[ 15w = 180 \]
  Next, divide both sides by 15:
  \[ w = \frac{180}{15} \]
  \[ w = 12 \]

✅ Solution: It will take Sarah 12 weeks to save enough money to buy the bicycle. 🚲

Let's solve the equation step-by-step to determine the number of solutions.

• Step 1: Apply the Distributive Property.
  Distribute the 3 on the left side.
  \[ 3x - 12 + 2x = 5x - 12 \]
• Step 2: Combine like terms on the left side.
  \[ (3x + 2x) - 12 = 5x - 12 \]
  \[ 5x - 12 = 5x - 12 \]
• Step 3: Move variable terms to one side.
  Subtract \(5x\) from both sides.
  \[ 5x - 5x - 12 = 5x - 5x - 12 \]
  \[ -12 = -12 \]

👉 When solving an equation, if you arrive at a statement that is always true (like \(-12 = -12\)), it means the equation is an identity. This implies that any real number for \(x\) will satisfy the equation.

✅ Solution: This equation has infinitely many solutions.

Let's use linear equations to find the dimensions of the rectangle.

• Step 1: Define variables.
  Let \(w\) be the width of the rectangle in cm.
  Let \(l\) be the length of the rectangle in cm.
• Step 2: Write equations based on the given information.
  We know the length is 6 cm more than the width:
  \[ l = w + 6 \quad \text{(Equation 1)} \]
  We know the perimeter \(P\) of a rectangle is \(2l + 2w\). The perimeter is 48 cm:
  \[ 2l + 2w = 48 \quad \text{(Equation 2)} \]
• Step 3: Substitute Equation 1 into Equation 2.
  Replace \(l\) in Equation 2 with \((w + 6)\):
  \[ 2(w + 6) + 2w = 48 \]
• Step 4: Solve the resulting linear equation for \(w\).
  Apply the distributive property:
  \[ 2w + 12 + 2w = 48 \]
  Combine like terms:
  \[ 4w + 12 = 48 \]
  Subtract 12 from both sides:
  \[ 4w = 48 - 12 \]
  \[ 4w = 36 \]
  Divide by 4:
  \[ w = \frac{36}{4} \]
  \[ w = 9 \]
  So, the width is 9 cm.
• Step 5: Find the length.
  Use Equation 1: \(l = w + 6\)
  Substitute \(w = 9\):
  \[ l = 9 + 6 \]
  \[ l = 15 \]
  So, the length is 15 cm.
• Step 6: Check your answer (optional but recommended!).
  Perimeter \( = 2(15) + 2(9) = 30 + 18 = 48 \text{ cm}\). This matches the given perimeter.

✅ Solution: The width of the rectangle is 9 cm and the length is 15 cm.