Math Problem Statement

Which function has the same y-intercept as the graph shown?

Solution

To determine which function has the same y-intercept as the graph shown, we need to find the y-intercept of the line in the image, then check each equation to see which one shares this y-intercept.

Step-by-Step Solution

  1. Identify the y-intercept from the graph: The graph shows a line crossing the y-axis. From visual inspection, it looks like the y-intercept is around y=3y = -3.

  2. Rewrite the equations in slope-intercept form (y = mx + b) if necessary to identify their y-intercepts:

    • (1) y=126x4y = \frac{12 - 6x}{4}: Simplify this to find y=33x2y = 3 - \frac{3x}{2}, which gives a y-intercept of 33.
    • (2) 27+3y=6x27 + 3y = 6x: Rearrange to solve for yy, resulting in y=2x9y = 2x - 9, so the y-intercept is 9-9.
    • (3) 6y+x=186y + x = 18: Rearrange to get y=16x+3y = -\frac{1}{6}x + 3, giving a y-intercept of 33.
    • (4) y+3=6xy + 3 = 6x: Simplify this to y=6x3y = 6x - 3, which gives a y-intercept of 3-3.

Conclusion

The correct answer is Option (4) y=6x3y = 6x - 3, as it has the same y-intercept of 3-3 as the graph.

Would you like more details on any of these steps?


Here are some follow-up questions that might deepen your understanding:

  1. How do you identify the y-intercept of a line from an equation?
  2. What are the steps to rearrange an equation to slope-intercept form?
  3. Why is it useful to express a line in slope-intercept form?
  4. What does the slope of a line represent in an equation?
  5. How can you determine the slope and y-intercept directly from a graph?

Tip: Always look for the y-intercept by setting x=0x = 0 in an equation when identifying it in non-standard forms.

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Math Problem Analysis

Mathematical Concepts

Algebra
Linear Equations
Y-Intercept
Slope-Intercept Form

Formulas

y = mx + b

Theorems

Slope-Intercept Form of a Linear Equation

Suitable Grade Level

Grades 8-10