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Chapter 3: Problem 102

An equation is \(y=-\frac{3}{4} x-9 .\) Find \(y\) when \(x=8\).

Short Answer

Expert verified
When \( x=8 \), \( y=-15 \).

Step by step solution

01

Understand the Equation

The given equation is in the form of a linear equation: \( y = -\frac{3}{4} x - 9 \). The goal is to find the value of \( y \) when \( x = 8 \).
02

Substitute the Value of x

Substitute \( x = 8 \) into the equation \( y = -\frac{3}{4} x - 9 \): \[ y = -\frac{3}{4} (8) - 9 \].
03

Multiply

Multiply \( -\frac{3}{4} \) by \( 8 \): \[ -\frac{3}{4} \times 8 = -6 \]. So the equation becomes: \[ y = -6 - 9 \].
04

Simplify

Simplify the expression: \[ -6 - 9 = -15 \]. Hence, \( y = -15 \).

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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 to solve equations by replacing one variable with a given value. This method is very straightforward, especially in linear equations. It involves simply 'substituting' the given value into the equation and then simplifying the resulted expression to find the desired value. In our exercise, we substituted the given value of x (which is 8) into the equation to find y. This substitution directly translates the equation from one involving two variables to an equation involving only one variable, making it much simpler to solve.
Linear Equation
A linear equation is an equation that makes a straight line when it is graphed on a coordinate plane. The general form of a linear equation in two variables is given by: \[ y = mx + c \] where `m` is the slope and `c` is the y-intercept. For example, in our given equation \[ y = -\frac{3}{4} x - 9 \] , the slope (m) is \[ -\frac{3}{4} \] and the y-intercept (c) is -9. Linear equations are foundational in algebra since they model relationships where the change between the variables is consistent.
Multiplication of Fractions
Multiplication of fractions might seem tricky at first, but it follows simple rules. You simply multiply the numerators together and the denominators together. For instance, in the exercise, we had to multiply \[ -\frac{3}{4} \] by 8. First, consider 8 as a fraction, which is \[ \frac{8}{1} \].

Now multiply:

\[ -\frac{3}{4} \times \frac{8}{1} = -\frac{3 \times 8}{4 \times 1} = -\frac{24}{4} = -6 \].

When multiplying fractions with whole numbers, it's useful to convert the whole number into a fraction to simplify the process. In this way, you can easily find the product.

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Most popular questions from this chapter

For exercises 95-98, 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 \(y\)-intercept of \(6 x+5 y=30\). Incorrect Answer: \(9 x+2 y=36\) $$ \begin{aligned} 9(0)+2 y &=36 \\ 2 y &=36 \\ \frac{2 y}{2} &=\frac{36}{2} \\ y &=18 \end{aligned} $$ The \(x\)-intercept is \((0,18)\).

(a) use a graphing calculator to graph \(y=\frac{3}{4} x-5\). Sketch the graph; describe the window. (b) for the given value of \(x\), use the Value command to find the value of \(y\). If it is necessary to change the window, describe the changed window. \(x=20\)

Explain why the slope of a vertical line is undefined.

(a) represent the information as two ordered pairs. (b) find the average rate of change, \(m\). The number of men enrolled in the fall in degreegranting institutions of higher education increased from \(7,575,000\) men in 2006 to \(8,770,000\) men in 2009 . Round to the nearest thousand. (Source: nces.ed.gov, 2011)

(a) find the \(y\)-intercept. (b) find the \(x\)-intercept. (c) use the slope formula to find the slope of the line. \(2 x+3 y=9\)
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