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Chapter 3: Problem 29

list (a) the coordinates of any \(y\) -intercept and (b) the coordinates of any \(x\) -intercept. Do not graph. $$ -4 x+3 y=150 $$

Short Answer

Expert verified
The y-intercept is at (0, 50) and the x-intercept is at (-37.5, 0).

Step by step solution

01

Isolate the y-variable to find the y-intercept

To find the y-intercept, set \( x = 0 \) in the equation \( -4x + 3y = 150 \). This simplifies to \( 3y = 150 \). Solve for \( y \) by dividing both sides by 3: \( y = 50 \). Therefore, the coordinates of the y-intercept are (0, 50).
02

Isolate the x-variable to find the x-intercept

To find the x-intercept, set \( y = 0 \) in the equation \( -4x + 3y = 150 \). This simplifies to \( -4x = 150 \). Solve for \( x \) by dividing both sides by -4: \( x = -37.5 \). Therefore, the coordinates of the x-intercept are (-37.5, 0).

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Key Concepts

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

Understanding the Y-Intercept
The y-intercept is a key concept in algebra. It represents the point where a line crosses the y-axis. To find the y-intercept, you set the value of x to zero in the linear equation. This isolates the y-variable and allows you to solve for it.
In our example, the linear equation is \(-4x + 3y = 150\). When we set x to 0, the equation simplifies to \(3y = 150\). Dividing both sides by 3, we find \(y = 50\). Thus, the y-intercept occurs at the coordinates (0, 50).
The y-intercept is crucial because it provides a specific point that can help in graphing the linear equation and understanding its behavior.
Understanding the X-Intercept
The x-intercept is another important aspect in algebra. It indicates the point where a line crosses the x-axis. To find the x-intercept, you set the value of y to zero in the linear equation. This isolates the x-variable and allows you to solve for it.
In our example, with the linear equation \(-4x + 3y = 150\), setting y to 0 simplifies it to \-4x = 150\. Dividing both sides by -4, we get \(x = -37.5\). Therefore, the x-intercept is at the coordinates (-37.5, 0).
The x-intercept is important for graphing and understanding where the line touches the x-axis.
What is a Linear Equation?
A linear equation is a mathematical expression that represents a straight line when graphed. It has the general format of \(ax + by = c\), where a, b, and c are constants. This form is called the standard form of a linear equation.
In our example, \(-4x + 3y = 150\) is a linear equation in standard form. Linear equations can be used to understand relationships between variables and to predict values based on input.
Key characteristics of linear equations include:
  • They graph as straight lines.
  • They have constant rates of change.
  • The highest power of any variable is 1.
Understanding linear equations helps build a strong foundation in algebra and is essential for solving more complex mathematical problems.

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Most popular questions from this chapter

Running. Zoe ran from the 4 -km mark to the 7 -km mark of a \(10-\mathrm{km}\) race in 15.5 min. At this rate, how long would it take Zoe to run a 5 -mi race? (Hint: \(\mathrm{km} \approx 0.62 \mathrm{mi} .)\)

Assume that \(r, p,\) and \(s\) are constants and that \(x\) and \(y\) are variables. Determine the slope and the y-intercept. $$r x+p y=s-r y$$

Find the function values. $$h(x)=3 x-2$$ a) \(h(4)\) b) \(h(8)\) c) \(h(-3)\) d) \(h(-4)\) e) \(h(a-1)\) f) \(h(a)-1\)

Find the function values. $$s(x)=\frac{3 x-4}{2 x+5}$$ a) \(s(10)\) b) \(s(2)\) c) \(s\left(\frac{1}{2}\right)\) d) \(s(-1)\) e) \(s(x+3)\)

Use a graphing calculator to find the function values. $$h(n)=8-n-\frac{1}{n}$$ a) \(h(0.2)\) b) \(h\left(-\frac{1}{4}\right)\)
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