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Chapter 3: Problem 87

How do we find the intercepts of the graph of an equation without having to graph the equation?

Short Answer

Expert verified
Substitute \( x = 0 \) to find the y-intercept and \( y = 0 \) to find the x-intercept.

Step by step solution

01

Identify the equation

First, identify the equation you are working with. For example, suppose we have a linear equation, like \( y = 2x + 3 \). We will use this equation to find the intercepts.
02

Find the y-intercept

The y-intercept occurs at the point where the graph crosses the y-axis. This happens when \( x = 0 \). Substitute \( x = 0 \) into the equation to find \( y \). For the equation \( y = 2x + 3 \), when \( x = 0 \), \( y = 2(0) + 3 = 3 \). Thus, the y-intercept is \( (0, 3) \).
03

Find the x-intercept

The x-intercept occurs at the point where the graph crosses the x-axis. This happens when \( y = 0 \). Substitute \( y = 0 \) into the equation to solve for \( x \). For \( y = 2x + 3 \), setting \( y = 0 \) gives \( 0 = 2x + 3 \). Solving for \( x \), we get \( 2x = -3 \) and \( x = -\frac{3}{2} \). Thus, the x-intercept is \( \left(-\frac{3}{2}, 0\right) \).

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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 type of equation that captures the relationship between two variables with a straight line when plotted on a graph. The simplest form of a linear equation is given as \( y = mx + b \), where \( m \) represents the slope of the line and \( b \) is the y-intercept. Linear equations are fundamental in algebra because they are easy to manipulate and understand.
  • Slope \( (m) \): This indicates how steep the line is. A positive slope means the line goes upward, while a negative slope indicates a downward direction.
  • Y-intercept \( (b) \): This is the point where the line crosses the y-axis. It shows the value of \( y \) when \( x \) is zero.
Linear equations are used to model many real-world situations, such as predicting trends or calculating rates of change. They are straightforward to solve, and understanding how they work is a crucial step in learning more advanced math topics.
Y-Intercept
Finding the y-intercept of a linear equation is quite simple. It's the point where the graph intersects the y-axis. For a linear equation \( y = mx + b \), the y-intercept is always \( (0, b) \). This means that when the variable \( x \) is set to zero, the resulting value of \( y \) gives you the y-intercept.
To find the y-intercept:
  • Set \( x = 0 \) in the equation.
  • Solve for \( y \), which will give you the y-value where the line crosses the y-axis.
In our example \( y = 2x + 3 \), substituting \( x = 0 \) gives \( y = 3 \). So, the y-intercept is \( (0, 3) \). This point is crucial in graphing as it provides a starting point for plotting the rest of the points on the graph accurately.
X-Intercept
The x-intercept is another vital feature of a linear graph. It is the point where the line crosses the x-axis, meaning the y-value at this point is zero. For the equation \( y = 2x + 3 \), to find the x-intercept:
  • Set \( y = 0 \) in the equation.
  • Solve for \( x \) to find the value where the line hits the x-axis.
For example, when you set \( y = 0 \) in the equation \( 0 = 2x + 3 \), it simplifies to \( 2x = -3 \) leading to \( x = -\frac{3}{2} \). Thus, the x-intercept is \( \left(-\frac{3}{2}, 0\right) \).
The x-intercept provides significant information as it shows where the function's output becomes zero. This can be important in various applications, such as economics, where it may represent a break-even point or solving zero-sum problems.

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Most popular questions from this chapter

For each pair of equations, determine whether their graphs are parallel, perpendicular, or neither. See Example 6 $$ \begin{aligned} &-4 x+3 y=-12\\\ &8 x+6 y=54 \end{aligned} $$

Find an equation of the line that passes through \((-6,3)\) and is perpendicular to the line \(y=-3 x-12 .\) Write the equation in slope-intercept form.

Explain what a politician meant when she said. "The speed at which the downtown area will be redeveloped is a function of the number of low-interest loans made available to the property owners."

On a quiz, a student was asked to find the slope of the graph of \(y=2 x+3 .\) She answered: \(m=2 x .\) Her instructor marked it wrong. Explain why the answer is incorrect.

Which of the following equations has the steeper graph, \(103 x-200 y=-400\) or \(17 x-33 y=-66 ?\)
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