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Magazine Chapter 2: Problem 70
Graph each equation \(4 x+3 y=0\)
Short Answer
Expert verified
The graph is a line passing through points (0,0) and (3,-4).
Step by step solution
01
Identify the Type of Equation
The given equation is \(4x + 3y = 0\), which is a linear equation in two variables. This represents a line on the Cartesian plane.
02
Rearrange to Slope-Intercept Form
Convert the equation \(4x + 3y = 0\) into the slope-intercept form \(y = mx + b\). Start by solving for \(y\):\[3y = -4x\]Divide both sides by 3 to isolate \(y\):\[y = -\frac{4}{3}x\]This is the slope-intercept form, where the slope \(m\) is \(-\frac{4}{3}\) and the y-intercept \(b\) is 0.
03
Determine Key Points for Plotting
Select key points by giving values to \(x\) and solving for \(y\). Choose \(x = 0\), then \(y = 0\), because the line passes through the origin.- When \(x = 0\), \(y = 0\) (Point: (0,0))Choose another point: Let \(x = 3\):- \(y = -\frac{4}{3}(3) = -4\) (Point: (3,-4))
04
Plot the Points on the Cartesian Plane
Plot the points (0,0) and (3,-4) on a graph. The point (0,0) is the origin, and (3,-4) is 3 units to the right and 4 units down from the origin.
05
Draw the Line
Using a ruler, draw a straight line through the plotted points (0,0) and (3,-4). This line represents the graph of the equation \(4x + 3y = 0\). Ensure the line extends in both directions indefinitely.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Linear Equation
A linear equation is an equation that models a straight line. It takes the form \( ax + by + c = 0 \), where \( a \), \( b \), and \( c \) are constants. In our scenario, the linear equation \( 4x + 3y = 0 \) represents a line on a two-dimensional plane. Linear equations can have either one variable (like \( ax + b = 0 \)) or two variables (like our example). When graphed, they create a line that can extend indefinitely in both directions.
- Deals with straight lines.
- Can be represented using two variables.
- Forms a straight, continuous line on a graph.
Slope-Intercept Form
The slope-intercept form of a linear equation is a way of expressing the equation of a line so that both its slope and intercept are immediately apparent. It's written as \( y = mx + b \). Here, \( m \) is the slope, representing the steepness and direction of the line, and \( b \) is the y-intercept, indicating where the line crosses the y-axis.
- Simplifies understanding of line characteristics.
- Provides quick insight into slope and intercept.
- Facilitates easy graph plotting.
Cartesian Plane
The Cartesian plane is a two-dimensional surface defined by two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical). It allows us to graphically represent mathematical concepts and equations such as the linear equation we have.
- Consists of two axes intersecting at the origin (0,0).
- Used to plot points, lines, and shapes.
- Each point is determined by a pair of coordinates \((x, y)\).
Coordinates
Coordinates are a set of values that show an exact position on the Cartesian plane. They are written as \((x, y)\), where \( x \) represents the horizontal position, and \( y \) represents the vertical position. These ordered pairs are essential in determining positions of points, which then help in plotting lines and graphs.
- Represent positions on the plane.
- Formed by x (horizontal) and y (vertical) values.
- Key in graphing linear equations.
