y=3x-7;y=-2x+8

y=3x-7;y=-2x+8

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Answer:

The equations y = 3x – 7 and y = -2x + 8 represent two linear equations in the form of y = mx + b, where m is the slope of the line and b is the y-intercept. Let’s break down these equations and understand their meanings:

  1. y = 3x – 7:
    • The slope of this line is 3, which means that for every increase of 1 unit in the x-coordinate, the y-coordinate increases by 3 units.
    • The y-intercept is -7, indicating that the line intersects the y-axis at the point (0, -7).
    • Example: If we substitute x = 2 into the equation, we can find the corresponding y-coordinate: y = 3(2) – 7 y = 6 – 7 y = -1 Therefore, when x = 2, y = -1, and the point (2, -1) lies on the line.
  2. y = -2x + 8:
    • The slope of this line is -2, meaning that for every increase of 1 unit in the x-coordinate, the y-coordinate decreases by 2 units.
    • The y-intercept is 8, indicating that the line intersects the y-axis at the point (0, 8).
    • Example: Let’s substitute x = -3 into the equation to find the corresponding y-coordinate: y = -2(-3) + 8 y = 6 + 8 y = 14 So, when x = -3, y = 14, and the point (-3, 14) lies on the line.

Now, let’s compare and analyze these two equations:

  1. Slope:
    • The slope of y = 3x – 7 is positive (3), indicating an upward slope.
    • The slope of y = -2x + 8 is negative (-2), indicating a downward slope.
  2. Y-intercept:
    • The y-intercept of y = 3x – 7 is -7, meaning the line intersects the y-axis at (0, -7).
    • The y-intercept of y = -2x + 8 is 8, indicating the line intersects the y-axis at (0, 8).

By comparing the slopes and y-intercepts, we can determine the following:

  1. Intersection Point:
    • To find the point where these two lines intersect, we need to solve the equations simultaneously. By equating the y-values, we get: 3x – 7 = -2x + 8 5x = 15 x = 3
    • Plugging the value of x back into either equation, we find: y = 3(3) – 7 y = 9 – 7 y = 2
    • Therefore, the lines y = 3x – 7 and y = -2x + 8 intersect at the point (3, 2).
  2. Relationship between the Lines:
    • The slope of y = 3x – 7 is greater than the slope of y = -2x + 8, which means the line y = 3x – 7 is steeper than y = -2x + 8.
    • Since the lines have different slopes, they will never be parallel.
    • The fact that the lines intersect confirms that they are not perpendicular either.

In summary, the equations y = 3x – 7 and y = -2x + 8 represent two lines with different slopes and y-intercepts. The line y = 3x – 7 has a positive slope and intersects the y-axis below the origin, while the line y = -2x + 8 has a negative slope and intersects the y-axis above the origin. These lines intersect at the point (3, 2).

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