Problem 20
Question
Graph the equations. $$ x+y=0 $$
Step-by-Step Solution
Verified Answer
Answer: The equation x + y = 0 is a linear equation. Two points that satisfy this equation are (0, 0) and (1, -1).
1Step 1: Identify the type of equation
This equation is a linear equation, written in the form Ax + By = C, where A = 1, B = 1, and C = 0.
2Step 2: Find two points that satisfy the equation
To draw a graph of a linear equation, we need to find at least two points that satisfy the equation. We can choose values for either x or y and solve the equation for the corresponding variable.
a) Choosing x = 0:
x + y = 0
0 + y = 0
y = 0
Point: (0, 0)
b) Choosing y = 0:
x + y = 0
x + 0 = 0
x = 0
Point: (0, 0)
Since the two points we found are the same, we need to choose a different value for x or y to find another point.
c) Choosing x = 1:
x + y = 0
1 + y = 0
y = -1
Point: (1, -1)
Now we have two different points: (0, 0) and (1, -1).
3Step 3: Draw the graph
Plot the two points (0, 0) and (1, -1) on a coordinate plane. Then, draw a straight line passing through these points. This line is the graph of the given equation x + y = 0.
Key Concepts
Linear EquationCoordinate PlanePlotting Points
Linear Equation
Understanding what a linear equation represents is fundamental in graphing it properly. A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable. Linear equations generate straight lines when graphed on a coordinate plane, hence the name 'linear'.
When we observe the equation from the exercise, \(x + y = 0\), it's in the standard form \(Ax + By = C\), where \(A\) and \(B\) are coefficients and \(C\) represents the constant term. These equations describe a relationship between two variables, typically \(x\) and \(y\). The solutions to a linear equation form a line, and every point on this line is a solution to the equation.
While some linear equations are more straightforward to graph because they are already in slope-intercept form (\(y=mx+b\)), the given equation necessitates finding specific points to plot, which requires algebraic manipulation if not already in a recognisable form.
When we observe the equation from the exercise, \(x + y = 0\), it's in the standard form \(Ax + By = C\), where \(A\) and \(B\) are coefficients and \(C\) represents the constant term. These equations describe a relationship between two variables, typically \(x\) and \(y\). The solutions to a linear equation form a line, and every point on this line is a solution to the equation.
While some linear equations are more straightforward to graph because they are already in slope-intercept form (\(y=mx+b\)), the given equation necessitates finding specific points to plot, which requires algebraic manipulation if not already in a recognisable form.
Coordinate Plane
The coordinate plane, also known as the Cartesian plane, is a two-dimensional surface defined by two number lines: the x-axis (horizontal) and the y-axis (vertical). These axes divide the plane into four quadrants, providing a framework for plotting points and visualizing equations.
When graphing linear equations, the coordinate plane allows us to display the relationship between \(x\) and \(y\) visually. Each point on the plane is determined by an \(x\) value (abscissa) and a \(y\) value (ordinate), written as an ordered pair \( (x, y) \). The origin, where both the x-axis and y-axis intersect, is the point \( (0, 0) \).
The importance of the coordinate plane lies in its ability to represent mathematical concepts spatially, such as the slope of a line or the intersection point of two lines, which often represent the solution to a set of linear equations.
When graphing linear equations, the coordinate plane allows us to display the relationship between \(x\) and \(y\) visually. Each point on the plane is determined by an \(x\) value (abscissa) and a \(y\) value (ordinate), written as an ordered pair \( (x, y) \). The origin, where both the x-axis and y-axis intersect, is the point \( (0, 0) \).
The importance of the coordinate plane lies in its ability to represent mathematical concepts spatially, such as the slope of a line or the intersection point of two lines, which often represent the solution to a set of linear equations.
Plotting Points
To accurately represent a linear equation graphically, we must be adept at plotting points. Plotting points involves identifying the exact location of points on the coordinate plane using their coordinates, and then drawing the linear equation by connecting these points with a straight line.
The process typically starts by assigning a value to one variable and solving for the other, obtaining an ordered pair as seen in the step-by-step solution for the equation \(x + y = 0\). Our first point is \( (0, 0) \), which is the intersection of the x and y axes. Using another value for \(x\), such as \(1\), we found the corresponding \(y\) value to be \( -1 \) since \(1 + y = 0 \rightarrow y = -1\), resulting in the point \( (1, -1) \).
It's crucial to plot more than one point because a single point does not determine a line. After the points are marked on the graph, a ruler is useful to draw a straight line through them, extending it across the grid, ensuring that the line represents all possible solutions to the equation.
The process typically starts by assigning a value to one variable and solving for the other, obtaining an ordered pair as seen in the step-by-step solution for the equation \(x + y = 0\). Our first point is \( (0, 0) \), which is the intersection of the x and y axes. Using another value for \(x\), such as \(1\), we found the corresponding \(y\) value to be \( -1 \) since \(1 + y = 0 \rightarrow y = -1\), resulting in the point \( (1, -1) \).
It's crucial to plot more than one point because a single point does not determine a line. After the points are marked on the graph, a ruler is useful to draw a straight line through them, extending it across the grid, ensuring that the line represents all possible solutions to the equation.
Other exercises in this chapter
Problem 20
For the following problems, write the equation of the line using the given information in slope-intercept form. $$ m=-\frac{3}{2}, y \text { -intercept }(0,0) $
View solution Problem 20
Graph the equation \(y=-3 x+2\).
View solution Problem 20
Parallel lines have the same slope.
View solution Problem 20
For the following problems, graph the equations. $$ 0 x+y=3 $$
View solution