Math Problem Statement

In Box #7, graph the equation: 4x + 7y = 28. Then answer the following question: Determine the x-intercept and y-intercept.

Solution

Let's solve the given equation step by step to address the questions in the image.

The equation provided is:

4x+7y=284x + 7y = 28

1. Determine the intercepts:

To find the x-intercept, set y=0y = 0 and solve for xx.

4x+7(0)=28    4x=28    x=284=74x + 7(0) = 28 \implies 4x = 28 \implies x = \frac{28}{4} = 7 Thus, the x-intercept is (7,0)(7, 0).

To find the y-intercept, set x=0x = 0 and solve for yy.

4(0)+7y=28    7y=28    y=287=44(0) + 7y = 28 \implies 7y = 28 \implies y = \frac{28}{7} = 4 Thus, the y-intercept is (0,4)(0, 4).

2. Check the other points:

Let’s see if the other points provided satisfy the equation.

  • For (0,7)(0, 7): 4(0)+7(7)=0+49=49(Not valid, since it does not equal 28).4(0) + 7(7) = 0 + 49 = 49 \quad (\text{Not valid, since it does not equal 28}).

  • For (4,0)(4, 0): 4(4)+7(0)=16+0=16(Not valid, since it does not equal 28).4(4) + 7(0) = 16 + 0 = 16 \quad (\text{Not valid, since it does not equal 28}).

  • For (7,4)(7, 4): 4(7)+7(4)=28+28=56(Not valid, since it does not equal 28).4(7) + 7(4) = 28 + 28 = 56 \quad (\text{Not valid, since it does not equal 28}).

Thus, the correct intercept points are (7, 0) and (0, 4), corresponding to the x-intercept and y-intercept.

Summary:

  • X-intercept: (7, 0)
  • Y-intercept: (0, 4)

Let me know if you need further details! Here are five related questions to deepen your understanding:

  1. How do you determine the slope of the line from the equation 4x+7y=284x + 7y = 28?
  2. How would you rewrite 4x+7y=284x + 7y = 28 in slope-intercept form?
  3. What is the slope of a line perpendicular to the line described by 4x+7y=284x + 7y = 28?
  4. If you were to translate the graph of 4x+7y=284x + 7y = 28 upwards by 2 units, how would the equation change?
  5. How would you graph the line if the equation were given in standard form Ax+By=CAx + By = C, like 4x+7y=284x + 7y = 28?

Tip: When solving for intercepts in linear equations, always set the other variable to 0 to simplify the process.

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Math Problem Analysis

Mathematical Concepts

Algebra
Linear Equations
Intercepts

Formulas

Standard form of a line: Ax + By = C
x-intercept: set y = 0
y-intercept: set x = 0

Theorems

Intercept theorem
Line equation

Suitable Grade Level

Grades 7-9