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Chapter 3: Problem 49

Evaluate \(y=6 x-8\) when \(x=0\).

Short Answer

Expert verified
When \( x = 0 \), \( y = -8 \).

Step by step solution

01

Understand the equation

The given equation is a linear function: \[ y = 6x - 8 \] It represents a straight line with a slope of 6 and a y-intercept of -8.
02

Substitute the value of x

To find the value of y when \( x = 0 \), substitute 0 into the equation for x: \[ y = 6(0) - 8 \]
03

Simplify the expression

Calculate the value of the expression by simplifying: \[ y = 0 - 8 \] \[ y = -8 \]
04

Write the final answer

After simplifying, the value of y when \( x = 0 \) is: \[ y = -8 \]

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Key Concepts

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

substitution method
The substitution method is a technique used in algebra to find the values of variables. It involves replacing one variable in an equation with a given value and simplifying the result. In our example, we started with the linear equation \( y = 6x - 8 \), and we wanted to find the value of y when x is 0. To do this, we substitute 0 for x in the equation.
By substituting, the equation becomes: \( y = 6(0) - 8 \).
This simplifies to \( y = -8 \). The substitution method helps us understand how the value of one variable affects another in the equation.
  • Always replace the variable with its given value first.
  • Carry out any multiplications or divisions next.
  • Finally, perform addition or subtraction to simplify.
linear functions
Linear functions represent relationships where changes in one variable result in proportional changes in another. These functions are usually written in the form \( y = mx + b \), where \m\ is the slope and \b\ is the y-intercept.
Our equation, \( y = 6x - 8 \), is a linear function. Here, the slope (m) is 6, indicating that for every one unit increase in x, y increases by 6 units.
Linear functions graph as straight lines and are straightforward to work with. They have important properties:
  • Constant slope: The slope, or rate of change, remains consistent.
  • Simplicity: Left one variable and perform algebraic operations to find the other variable.
  • Predictability: Easy to predict future values within the function.
y-intercept
The y-intercept is the point where the graph of an equation crosses the y-axis. In the context of a linear function, it鈥檚 represented by \b\ in the equation \( y = mx + b \). For the given linear equation \( y = 6x - 8 \), the y-intercept is -8.
This means that when x is 0, y is -8. The point (0, -8) is where the line cuts across the y-axis.
Identifying the y-intercept is crucial because:
  • It provides a starting point for graphing the line.
  • Helps in understanding the relationship between the variables at the origin (where x=0).
  • Makes it easier to visualize how the linear function will look when plotted.

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Most popular questions from this chapter

(a) find the \(y\)-intercept. (b) find the \(x\)-intercept. (c) use the slope formula to find the slope of the line. \(3 x+10 y=60\)

Use the slope formula to find the slope of the line that passes through the points. \((5,10) ;(7,16)\)

Use the slope formula to find the slope of the line that passes through the points. \((-5,4) ;(-9,-3)\)

For exercises 89-92, the completed problem has one mistake. (a) Describe the mistake in words, or copy down the whole problem and highlight or circle the mistake. (b) Do the problem correctly. Problem: Find the slope of the line that passes through \((7,1)\) and \((9,4)\). Incorrect Answer: \(m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\) $$ \begin{aligned} m &=\frac{9-7}{4-1} \\ m &=\frac{2}{3} \end{aligned} $$

Use the slope formula to find the slope of the line that passes through the points. \(\left(\frac{1}{3}, \frac{9}{7}\right) ;\left(\frac{5}{3}, \frac{3}{7}\right)\)
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