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Chapter 11: Problem 57

Find the \(x\) - and \(y\) -intercepts of the graph of \(4 x+3 y=12\)

Short Answer

Expert verified
x-intercept: 3, y-intercept: 4

Step by step solution

01

Find the x-intercept

To find the x-intercept, set \(y = 0\) in the given equation \(4x + 3y = 12\). This gives us \(4x + 3(0) = 12\). Simplify this to \(4x = 12\). Then, solve for \(x\) by dividing both sides by 4: \(x = \frac{12}{4} = 3\). Thus, the x-intercept is \(3\).
02

Find the y-intercept

To find the y-intercept, set \(x = 0\) in the given equation \(4x + 3y = 12\). This gives us \(4(0) + 3y = 12\). Simplify this to \(3y = 12\). Then, solve for \(y\) by dividing both sides by 3: \(y = \frac{12}{3} = 4\). Thus, the y-intercept is \(4\).

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Key Concepts

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

Finding the x-intercept
The x-intercept is the point where the line crosses the x-axis. At this point, the value of y is always zero.
To find the x-intercept of the equation, follow these steps:
  • Set y to 0 in the given equation. For example, with the equation 4x + 3y = 12, it becomes 4x + 3(0) = 12.
  • Simplify the equation. Here, it results in 4x = 12.
  • Solve for x by dividing both sides by 4. This gives x = 3.
So, the x-intercept is 3. This means the line crosses the x-axis at the point (3, 0).
Finding the y-intercept
The y-intercept is the point where the line crosses the y-axis. At this point, the value of x is always zero.
To find the y-intercept of the equation, follow these steps:
  • Set x to 0 in the given equation. For example, with the equation 4x + 3y = 12, it becomes 4(0) + 3y = 12.
  • Simplify the equation. Here, it results in 3y = 12.
  • Solve for y by dividing both sides by 3. This gives y = 4.
So, the y-intercept is 4. This means the line crosses the y-axis at the point (0, 4).
Solving linear equations
Solving linear equations involves finding the value of the unknown variable that makes the equation true. Here鈥檚 a basic approach to solve most linear equations:
  • Isolate the variable on one side of the equation. To do this, you may need to add, subtract, multiply, or divide both sides of the equation by the same number.
  • Simplify both sides of the equation as much as possible. Combine like terms and reduce fractions when necessary.
  • Check your solution by plugging it back into the original equation to ensure it satisfies the equation.
In the examples above:
  • For finding the x-intercept, we set y = 0 and solved for x.
  • For finding the y-intercept, we set x = 0 and solved for y.
Remember, linear equations form straight lines when graphed, and the intercepts help to understand where these lines intersect the axes.

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Most popular questions from this chapter

Graph each ellipse. $$ \frac{x^{2}}{9}+\frac{y^{2}}{16}=1 $$

A lithotripter is a machine used to crush kidney stones using shock waves. The patient is placed in an elliptical tub with the kidney stone at one focus of the ellipse. A beam is projected from the other focus to the tub, so that it reflects to hit the kidney stone. Suppose a lithotripter is based on the ellipse with equation $$ \frac{x^{2}}{36}+\frac{y^{2}}{9}=1 $$ How far from the center of the ellipse must the kidney stone and the source of the beam be placed? (Hint: Use the fact that \(c^{2}=a^{2}-b^{2},\) since \(a>b\) here.

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