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Chapter 5: Problem 13

Find the \(x\) - and \(y\) -intercepts of the equation. $$2 x+3 y=12$$

Short Answer

Expert verified
The \(x\)-intercept is (6, 0) and the \(y\)-intercept is (0, 4).

Step by step solution

01

Identify the equation

The given equation is \(2x + 3y = 12\). This is a linear equation in two variables.
02

Find the x-intercept

To find the \(x\)-intercept, set \(y = 0\) and solve for \(x\).\[2x + 3(0) = 12\]\[2x = 12\]\[x = 6\]So, the \(x\)-intercept is \( (6, 0) \).
03

Find the y-intercept

To find the \(y\)-intercept, set \(x = 0\) and solve for \(y\).\[2(0) + 3y = 12\]\[3y = 12\]\[y = 4\]So, the \(y\)-intercept is \( (0, 4) \).

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Key Concepts

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

x-intercept
Understanding the x-intercept of a linear equation is easy and important. You are essentially finding the point where the line crosses the x-axis. At the x-intercept, the value of y is always 0.
To find the x-intercept for the given equation, follow these steps:
  • Set y to 0 in the equation.
  • Solve the resulting equation for x.
For example, in the equation: \(2x + 3y = 12\), set y to 0:
\[2x + 3(0) = 12\]

Solving this, we get:
\[2x = 12\]
\[x = 6\]
Therefore, the x-intercept is (6, 0). This tells you that at the point where the line crosses the x-axis, the x-coordinate is 6.
y-intercept
The y-intercept is where the line crosses the y-axis. Here, the value of x is always 0. Finding the y-intercept involves a similar process to finding the x-intercept.
Follow these steps to find the y-intercept for any linear equation:
  • Set x to 0 in the equation.
  • Solve the resulting equation for y.
Using the given equation: \(2x + 3y = 12\), set x to 0:
\[2(0) + 3y = 12\]

Simplify and solve for y:
\[3y = 12\]
\[y = 4\]
Therefore, the y-intercept is (0, 4). This means that the point where the line crosses the y-axis has a y-coordinate of 4.
solving linear equations
Solving linear equations is a fundamental skill in algebra. A linear equation is any equation that can be written in the form \(Ax + By = C\), where A, B, and C are constants.
To solve a linear equation, you need to isolate the variable of interest. This often involves multiple steps:
  • Combine like terms.
  • Use addition or subtraction to move terms from one side of the equation to the other.
  • Use multiplication or division to solve for the variable.
Let's break down the process using the given equation \(2x + 3y = 12\):

To find the x-intercept, set y to 0.
This gives \[2x = 12\] then solving for x, \[x = 6\]

To find the y-intercept, set x to 0.
This gives \[3y = 12\] then solving for y, \[y = 4\]

Remember, the key steps include:
  • Substitute known values.
  • Simplify to isolate the variable.
  • Check your work by substituting the solution back into the original equation.
With practice, solving linear equations becomes second nature!

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Most popular questions from this chapter

Round off to the nearest hundredth when necessary. In physiology a jogger's heart rate \(N\), in beats per minute, is related linearly to the jogger's speed \(s .\) A certain jogger's heart rate is 80 beats per minute at a speed of \(15 \mathrm{ft} / \mathrm{sec}\) and 82 beats per minute at a speed of \(18 \mathrm{ft} / \mathrm{sec} .\) (a) Write an equation relating the jogger's speed and heart rate. (b) Predict this jogger's heart rate if she jogs at a speed of \(20 \mathrm{ft} / \mathrm{sec}\). (c) According to the equation obtained in part (a), what is the jogger's heart rate at rest? [Hint: At rest the jogger's speed is 0.]

Write an equation of the line satisfying the given conditions. Passing through \((0,5)\) and \((5,2)\)

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Round off to the nearest hundredth when necessary. A sidewalk food vendor knows that selling 80 franks costs a total of 73 dollar and selling 100 franks costs a total of 80 dollar Assume that the total cost \(c\) of the franks is linearly related to the number \(f\) of franks sold. (a) Write an equation relating the total cost of the franks to the number of franks sold. (b) Find the cost of selling 90 franks. (c) How many franks can be sold for a total cost of 90.50 dollar (d) The costs that exist even if no items are sold are called the fixed costs. Find the vendor's fixed costs. I Hint: If no franks are sold then \(f=0.1\)

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