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Chapter 5: Problem 40

Solve the given problems. Find the intercepts of the line \(y=m x+b\).

Short Answer

Expert verified
Y-intercept: \((0, b)\); X-intercept: \((-\frac{b}{m}, 0)\).

Step by step solution

01

Understanding the Intercepts

The intercepts of a line are the points where the line crosses the axes. The line crosses the y-axis at the y-intercept and the x-axis at the x-intercept.
02

Finding the Y-Intercept

To find the y-intercept, set the value of \(x\) to zero in the equation \(y = mx + b\). This simplifies the equation to \(y = b\). Therefore, the y-intercept is the point \((0, b)\).
03

Finding the X-Intercept

To find the x-intercept, set the value of \(y\) to zero in the equation \(y = mx + b\). This gives us \(0 = mx + b\). Solving for \(x\), we get \(x = -\frac{b}{m}\). Therefore, the x-intercept is the point \((-\frac{b}{m}, 0)\).

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Key Concepts

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

Intercepts
Intercepts are the points where a graph meets the axes on a coordinate plane. These points are crucial for understanding the behavior and position of a line, especially in linear equations.
  • Y-Intercept: The point where the line crosses the y-axis.
  • X-Intercept: The point where the line crosses the x-axis.
Intercepts help visualize the line and can be a starting point for graphing linear equations. By pinpointing these intercepts, you understand how the line interacts with the axes.
Y-Intercept
The y-intercept of a line is the point where it crosses the y-axis. This point is found by substituting the value of zero for the variable x in the equation of the line, which is represented as \( y = mx + b \).When you set \( x = 0 \), the equation simplifies down to \( y = b \).This tells us that the y-intercept occurs at the point \( (0, b) \).
  • Understanding: The y-intercept is the starting point of the line if you imagine the line moving across the graph from the y-axis.
  • Example: If the equation is \( y = 4x + 3 \), then the y-intercept is \((0, 3)\).
In linear equations, the y-intercept is an especially handy component when plotting the graph or for comparing different linear relationships.
X-Intercept
The x-intercept is where the line crosses the x-axis. To identify this point, replace the variable y with zero in the line's equation \( y = mx + b \).This modifies the equation to \( 0 = mx + b \).
  • Solve for \( x \) to find \( x = -\frac{b}{m} \).
  • The x-intercept is thus represented by the point \( \left(-\frac{b}{m}, 0\right) \).
The x-intercept helps demonstrate how far and in which direction the line extends along the x-axis. In graphing, it ensures that the line meets the horizontal axis accurately, which can profoundly influence the overall slope perception.
Equation of a Line
An equation of a line, often written as \( y = mx + b \), is a mathematical statement that describes the relationship between the x and y coordinates of any point on that line. Here, the letter \( m \) signifies the slope of the line, and \( b \) represents the y-intercept.
This equation is a fundamental tool in algebra. It provides a concise way to express a line's slope and intercept.
  • Slope (m): Indicates the steepness and direction of the line.
  • Intercept (b): Shows where the line touches the y-axis.
Understanding this equation enables you to predict y-values for given x-values or vice versa. It's a versatile format that's key in graphing and interpreting linear data.

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Most popular questions from this chapter

Solve the given systems of equations by the method of elimination by substitution. $$\begin{aligned} &33 x+2 y=34\\\ &40 y=9 x+11 \end{aligned}$$

Solve the given systems of equations by either method of this section. $$\begin{aligned} &60 x-40 y=80\\\ &2.9 x-2.0 y=8.0 \end{aligned}$$

In order to make the coefficients easier to work with, first multiply each term of the equation or divide each term of the equation by a number selected by inspection. Then proceed with the solution of the system by an appropriate algebraic method. $$\begin{array}{l} 250 R+225 Z=400 \\ 375 R-675 Z=325 \end{array}$$

Solve the indicated or given systems of equations by an appropriate algebraic method. A \(6.0 \%\) solution and a \(15.0 \%\) solution of a drug are added to \(200 \mathrm{mL}\) of a \(20.0 \%\) solution to make \(1200 \mathrm{mL}\) of a \(12.0 \%\) solution for a proper dosage. The equations relating the number of milliliters of the added solutions are \(x+y+200=1200\) \(0.060 x+0.150 y+0.200(200)=0.120(1200)\) Find \(x\) and \(y\) (to three significant digits).

Solve the given systems of equations by either method of this section. $$\begin{aligned} &66 x+66 y=-77\\\ &33 x-132 y=143 \end{aligned}$$
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