Math Problem Statement

If the line through (-2, 4) and (5, d) is parallel to y = 3x + 4, what is the value of d?

Solution

Here's a clearer breakdown of the problems from your image:

  1. Problem 16: If the line through points (2,4)(-2, 4) and (5,d)(5, d) is parallel to the line y=3x+4y = 3x + 4, what is the value of dd?

  2. Problem 17: Solve the equation 5(x+4)2+7=37\frac{5(x + 4)}{2} + 7 = 37 and then evaluate 3x93x - 9.

  3. Problem 18: Solve the equation 8b(9)10=17\frac{-8b - (-9)}{-10} = 17.

I will now solve each of these problems step by step.

Problem 16: Equation of the Line and Value of dd

Given that the slope of the line y=3x+4y = 3x + 4 is 3, we find the slope between the points (2,4)(-2, 4) and (5,d)(5, d). The slope formula between these points is: m=d45(2)=d47m = \frac{d - 4}{5 - (-2)} = \frac{d - 4}{7}

Setting this equal to 3: d47=3\frac{d - 4}{7} = 3

Solving for dd: d4=21    d=25d - 4 = 21 \implies d = 25

Problem 17: Solving the Equation

Given: 5(x+4)2+7=37\frac{5(x + 4)}{2} + 7 = 37

Subtract 7 from both sides: 5(x+4)2=30\frac{5(x + 4)}{2} = 30

Multiply both sides by 2: 5(x+4)=605(x + 4) = 60

Divide both sides by 5: x+4=12x + 4 = 12

Subtract 4 from both sides: x=8x = 8

Now, evaluate 3x93x - 9: 3(8)9=249=153(8) - 9 = 24 - 9 = 15

Problem 18: Solving for bb

Given: 8b(9)10=17\frac{-8b - (-9)}{-10} = 17

Simplify the numerator: 8b+910=17\frac{-8b + 9}{-10} = 17

Multiply both sides by 10-10: 8b+9=170-8b + 9 = -170

Subtract 9 from both sides: 8b=179-8b = -179

Divide both sides by 8-8: b=1798b = \frac{179}{8}

Simplified, b=22.375b = 22.375.


If you have any questions about these solutions or need further clarification, please let me know!

Additional Questions to Expand Your Understanding:

  1. How do you determine the slope between two points on a graph?
  2. What properties do parallel lines share regarding their slopes?
  3. How can you solve multi-step equations that involve fractions?
  4. What are the common mistakes to avoid when solving for variables in equations?
  5. How can you double-check your solutions for equations?

Tip: Always check if your final answers make sense by substituting them back into the original equations.

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Math Problem Analysis

Mathematical Concepts

Algebra
Slope
Parallel Lines

Formulas

Slope formula m = (y2 - y1) / (x2 - x1)

Theorems

Parallel line slopes are equal

Suitable Grade Level

Grade 9