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Chapter 1: Problem 24

Find the \(x\) - and \(y\) -intercepts of the graph of the equation. $$y=8-3 x$$

Short Answer

Expert verified
The \(x\)-intercept for the given equation is \(x=8/3\) and the \(y\)-intercept is \(y = 8\).

Step by step solution

01

Finding the \(x\)-intercept

To find the \(x\)-intercept, we set \(y = 0\) in the equation \(y = 8 - 3x\), and solve for \(x\). Doing this gives:0 = 8 - 3xThis simplifies to 3x = 8So, x = 8/3.
02

Finding the \(y\)-intercept

For the \(y\)-intercept, we set \(x = 0\) in the initial equation, which is \(y = 8 - 3x\), and solve for \(y\). Therefore:y = 8 - 3(0)Hence, y = 8.

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Key Concepts

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

X-Intercept Calculation
Understanding how to find the x-intercept of a graph is crucial in algebra. The x-intercept is where the line crosses the x-axis, which occurs when the value of y is zero. To find it, you replace y with 0 in the equation and solve for x. For instance, in the equation y = 8 - 3x, setting y to zero yields 0 = 8 - 3x. Solving for x, we get x = 8/3. This means the graph crosses the x-axis at the point (8/3, 0).

When practicing x-intercept calculation:
  • Always set y to zero because the y-coordinate of any point on the x-axis is zero.
  • Isolate x and solve the resulting equation to find the x-intercept.
Y-Intercept Calculation
The y-intercept is found where a line crosses the y-axis, and at this point, x is always zero. For the equation \(y = 8 - 3x\), we set x to zero and solve for y, making the equation \(y = 8 - 3(0)\), which simplifies to \(y = 8\). Therefore, the line intersects the y-axis at \(0, 8\).

To correctly determine the y-intercept:
  • Set x to zero in the equation and solve for y.
  • The resulting value is the y-coordinate of the y-intercept, while the x-coordinate will be zero.
Linear Equations
Linear equations form the backbone of algebra and represent straight lines on a graph. They are written in the form \(y = mx + b\), where \(m\) is the slope, and \(b\) is the y-intercept. In our example, \(y = 8 - 3x\), \(m = -3\) and \(b = 8\). These equations are linear because, no matter the value of x, y will always be a straight line.

In analyzing linear equations, remember:
  • The slope \(m\) indicates the steepness and direction of the line.
  • The y-intercept \(b\) shows where the line crosses the y-axis.
Solving Equations
When we talk about solving equations, it means finding the value for the variable that makes the equation true. To solve a linear equation like \(y = 8 - 3x\), you perform algebraic operations like addition and subtraction to isolate the variable on one side of the equation. From our calculations:
  • To solve for x (x-intercept), set y to 0 and isolate x.
  • To solve for y (y-intercept), set x to 0 and solve for y, which in this case is straightforward: \(y = 8\).

Always apply the inverse operations to both sides of the equation to maintain the equality. For example, if you subtract by a number on one side, do the same on the other side. These steps ensure that you isolate the variable and find the solution.

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Most popular questions from this chapter

Use Hooke's Law for springs, which states that the distance a spring is stretched (or compressed) varies directly as the force on the spring. A force of 265 newtons stretches a spring 0.15 meter. (a) What force is required to stretch the spring 0.1 meter? (b) How far will a force of 90 newtons stretch the spring?

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The simple interest on an investment is directly proportional to the amount of the investment. An investment of \(\$ 6500\) will earn \(\$ 211.25\) after 1 year. Find a mathematical model that gives the interest \(I\) after 1 year in terms of the amount invested \(P\).

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The diameter of the largest particle that can be moved by a stream varies approximately directly as the square of the velocity of the stream. A stream with a velocity of \(\frac{1}{4}\) mile per hour can move coarse sand particles about 0.02 inch in diameter. Approximate the velocity required to carry particles 0.12 inch in diameter.
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