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Chapter 3: Problem 75

Graph equation. Solve for \(y\) first, when necessary. \(7 x-y=1\)

Short Answer

Expert verified
Solution: The equation solved for \(y\) is \(y = 7x - 1\).

Step by step solution

01

Isolate the term with y

Start by moving terms not involving \(y\) to the other side of the equation. Given equation is \(7x - y = 1\). Move \(7x\) to the right side to isolate the \(y\)-term. This gives us \(-y = -7x + 1\).
02

Solve for y

To solve for \(y\), we need it to be positive. Multiply both sides by -1, which gives \(y = 7x - 1\). Now, the equation is in the form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.

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Key Concepts

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

Slope-Intercept Form
The slope-intercept form of a linear equation is one of the most popular ways to express a line. It is written as \(y = mx + b\). In this format:
  • \(m\) represents the slope of the line. This describes how steep the line is and the direction it goes. A positive \(m\) means the line goes upwards from left to right, while a negative \(m\) means it goes downwards.
  • \(b\) is the y-intercept. This is the point where the line crosses the y-axis. It's the value of \(y\) when \(x\) is zero.
To convert any linear equation into the slope-intercept form, ensure that \(y\) is isolated on one side of the equation. In our given problem, the equation \(y = 7x - 1\) is already in this form. Here, the slope \(m\) is 7 and the y-intercept \(b\) is \(-1\). Understanding this form is crucial, as it makes graphing equations straightforward.
Graphing Equations
Graphing a linear equation involves drawing a straight line on a coordinate plane. Once you have the equation in the slope-intercept form \(y = mx + b\), you can start plotting. Here's how:
  • First, plot the y-intercept \(b\). For the equation \(y = 7x - 1\), this point is (0, -1).
  • Next, use the slope \(m\) to find another point. The slope tells you how to move from the y-intercept. With a slope of 7, it means you go up 7 units and 1 unit to the right for each step along the x-axis.
  • Draw a line through these points, extending it across the grid. This line is the graphical representation of the equation.
By following these steps, you turn a basic equation into a visual graph. Using the slope helps identify the line's direction and steepness, while the y-intercept shows its initial position on the axis.
Solving for Variables
Solving for a variable means rearranging an equation so that the variable stands alone on one side. In a linear equation with \(x\) and \(y\), we often solve for \(y\) to convert it into the slope-intercept form. Here's the process:
  • Identify the term with \(y\) and isolate it on one side, as shown in the exercise.
  • In \(7x - y = 1\), move \(7x\) to the other side, resulting in \(-y = -7x + 1\).
  • Remove any negative or unnecessary factors by multiplying or dividing. Multiplying by \(-1\), we solve for \(y\) giving \(y = 7x - 1\).
Having a clear pathway to isolate and solve for variables leads to easier manipulation of equations. This process is fundamental, especially when preparing the equation for graphing. It ensures clarity on how a change in one variable affects the other within an equation.

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Most popular questions from this chapter

Find the slope of a line perpendicular to the line passing through the given two points. See Example \(9 .\) \((-7,6)\) and \((0,4)\)

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Explain the error in the following solution. $$ \begin{array}{l} {\text { If } f(x)=x^{2}+7 x+1, \text { find } f(10)} \\ {\qquad \begin{aligned} \widehat{f}(10) &=10^{2}+7 x+1 \\ &=100+7 x+1 \\ &=101+7 x \end{aligned}} \end{array} $$

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