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Chapter 1: Problem 62

Find the slope and \(y\)-intercept of the line, and draw its graph. \(3 x-4 y=12\)

Short Answer

Expert verified
Slope is \(\frac{3}{4}\), y-intercept is \(-3\), line goes through points (0, -3) and (4, 0).

Step by step solution

01

Put the equation in slope-intercept form

To find the slope and the y-intercept of the line, we need to express the equation in the slope-intercept form, which is \(y = mx + b\). Start with the given equation \(3x - 4y = 12\). Solve for \(y\):\[-4y = -3x + 12\]Next, divide every term by \(-4\):\[y = \frac{3}{4}x - 3\]Now, the equation is in the form \(y = mx + b\), with \(m = \frac{3}{4}\) and \(b = -3\).
02

Identify the slope and y-intercept

From the equation \(y = \frac{3}{4}x - 3\), identify the slope \(m\) as \(\frac{3}{4}\) and the y-intercept \(b\) as \(-3\). This means the line rises 3 units for every 4 units it moves to the right, and it crosses the y-axis at \(y = -3\).
03

Plot the y-intercept

To draw the graph, start by plotting the y-intercept \((0, -3)\) on the graph. This is the point where the line crosses the y-axis.
04

Use the slope to find another point

From the y-intercept \((0, -3)\), use the slope \(\frac{3}{4}\) which means rising 3 units and running 4 units to the right. Starting from \((0, -3)\), move up 3 units to \(y = 0\) and 4 units to the right to \(x = 4\), giving the point \((4, 0)\).
05

Draw the line

Using the points \((0, -3)\) and \((4, 0)\), draw the line that passes through both on the graph. Extend the line across the graph to represent the solution of the linear equation.

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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 an essential tool for quickly understanding the behavior of a line. It is expressed as \(y = mx + b\), where \(m\) represents the slope and \(b\) is the y-intercept. By converting an equation to this format, you can easily identify these two key characteristics.
  • Slope (\(m\)): Indicates how steep the line is. It describes the rate of change, or how much \(y\) changes for a change in \(x\).
  • Y-intercept (\(b\)): The y-coordinate where the line crosses the y-axis. It's the value of \(y\) when \(x = 0\).
The original equation, \(3x - 4y = 12\), transformed into slope-intercept form by isolating \(y\), becomes \(y = \frac{3}{4}x - 3\). Here, the slope is \(\frac{3}{4}\) and the y-intercept is \(-3\). This format makes it simple to both visualize the line and analyze its features.
Graphing linear equations
Once you have an equation in slope-intercept form, graphing it becomes a straightforward task, which provides a visual understanding of how the line behaves. To begin graphing:
  • Plot the y-intercept: Start by marking the point \((0, b)\) on the y-axis. This point is given by the y-intercept, which in our example is \((0, -3)\).
  • Use the slope: From the y-intercept, use the slope \(\frac{3}{4}\) to find another point. The slope tells us to rise 3 units for every 4 units we move to the right.
  • Draw the line: Connect the y-intercept to the new point, and extend the line in both directions. This line is the graph of the equation \(y = \frac{3}{4}x - 3\).
Graphing helps see how the line interacts with the axes and nearby points. It's a visual representation that displays the relationship between \(x\) and \(y\). It simplifies understanding complex relationships through a clear and straightforward depiction.
Linear equations
Linear equations are equations of the first degree, which means they have no exponents higher than one. They represent straight lines when graphed on a coordinate plane. A linear equation typically has the general form:
  • Standard form: \(Ax + By = C\), where \(A\), \(B\), and \(C\) are constants, and both \(A\) and \(B\) are not zero.
  • Slope-intercept form: \(y = mx + b\), helpful for quickly determining the line's characteristics.
The transformation of a standard form equation into slope-intercept form, as shown with \(3x - 4y = 12\), is a strategic step for graphing and analysis. With just a glance, we can see the change in the y-value relative to the x-value (the slope), and where the line crosses the y-axis (the y-intercept). Linear equations model relationships where changes between variables are constant, such as speed over time or cost per item.

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