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Chapter 11: Problem 15

Find the \(y\) intercept for \(3 x-7 y=21\) A. \((7,0)\) B. \((-3,0)\) \(C .(0,-3)\) D. \((-7,0)\)

Short Answer

Expert verified
The y-intercept is (0, -3), which matches option C.

Step by step solution

01

Understanding the Objective

The goal is to find the y-intercept of the given linear equation. The y-intercept is the point at which the line crosses the y-axis, and it has coordinates (0, b), where b is the value of y when x is zero.
02

Setting x to Zero

To find the y-intercept, we set the value of x to zero in the equation. So, start with the equation: \(3x - 7y = 21\). Set \(x = 0\), then the equation simplifies to \(-7y = 21\).
03

Solving for y

Now, solve for y in the equation \(-7y = 21\) by dividing both sides of the equation by -7:\[ y = \frac{21}{-7} = -3 \].
04

Identifying the y-intercept

Now that we've solved for y, the y-intercept is the point where x is 0 and y is -3. Therefore, the y-intercept can be expressed as point (0, -3).
05

Matching with Given Options

Match the calculated y-intercept (0, -3) to one of the given options. It corresponds to option C: \((0, -3)\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Linear Equation
A linear equation is a mathematical expression that represents a straight line on a graph. Typically, it is written in the form \(ax + by = c\) or in the slope-intercept form \(y = mx + b\), where:
  • \(a\), \(b\), and \(c\) are constants,
  • \(x\) and \(y\) are variables representing coordinates on the Cartesian plane,
  • \(m\) is the slope of the line,
  • \(b\) is the y-intercept.
To identify a linear equation, look for these defining traits: a maximum of two variables with no exponents or roots. These equations are fundamental in modeling situations where there is a constant rate of change between variables. Understanding how to manipulate them is key to solving diverse mathematical and real-world problems.
Coordinates
Coordinates form the foundation for graphing equations and understanding geometric relationships. A "coordinate" in the context of the Cartesian plane refers to a set of values that pinpoint a specific location. The pair is represented as \((x, y)\):
  • \(x\): Horizontal position or distance from the y-axis
  • \(y\): Vertical position or distance from the x-axis
In the problem, the y-intercept is a point on the y-axis where the line crosses. It will always have an \(x\) value of zero because it is directly on the y-axis. Understanding coordinates is crucial for graphing as they visually represent equations on the plane, allowing us to analyze and interpret mathematical relationships easily.
Solving Equations
Solving equations involves finding the value of unknown variables that satisfy the equation. In this exercise, our task is to find the y-intercept by setting \(x = 0\) and solving for \(y\). The step-by-step process is:
  • Substitute \(x = 0\) into the equation \(3x - 7y = 21\).
  • This simplifies to \(-7y = 21\).
  • Divide both sides by \(-7\) to isolate \(y\): \(y = \frac{21}{-7}\).
  • Simplify to find \(y = -3\).
This approach follows standard solving techniques, ensuring accurate results. Such skills are essential for handling more complex problems and equations across various fields of study.
Graphing
Graphing is a powerful visual tool in mathematics that allows us to see the solutions of equations and their relationships spatially. By plotting equations on a Cartesian plane, we observe intersections and relationships between lines. Here's a simple rundown of how to graph the given linear equation:
  • Identify the y-intercept as \((0, -3)\).
  • Plot this point on the y-axis of the graph.
  • To complete the graph, find another point by choosing a value for \(x\), say \(x = 1\), and solve for \(y\). This helps in determining the slope visually.
  • Once a second point is plotted, draw a line through both points extending across the graph.
Graphing not only helps in solving equations visually but also aids in understanding the slope and y-intercept visually, which gives a clearer depiction of linear relationships.

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Most popular questions from this chapter

Which is a solution to \(4 x-y=8 ?\) A. \((2,2)\) B. \((-2,0)\) C. \((3,-4)\) D. \((0,-8)\)

Find \(x\) when \(y=-3\) for \(x+3 y=-10\) A. \(x=1\) B. \(x=2\) \(C . x=-1\) D. \(x=-2\)

The slope of a horizontal line is A. 0 B. 1 C. undefined \(D,-1\)

Find \(y\) when \(x=-3\) for \(4 x+5 y=-12\) A. 5 B. 0 c. \(-2\) \(D .-3\)

Find the slope of a line containing the two points whose coordinates are \((5,7)\) and \((-2,2)\) A. \(\frac{7}{5}\) B. \(-\frac{5}{7}\) c. \(-\frac{7}{5}\) D. \(\frac{5}{7}\)
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