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Chapter 4: Problem 33

Solve by graphing. $$ \begin{aligned} &y=2 x+5 \\ &y=-3 x+5 \end{aligned} $$

Short Answer

Expert verified
The solution is \( x = 0, y = 5 \).

Step by step solution

01

Plot the first equation

Start by plotting the equation \(y = 2x + 5\). Choose a few values for \(x\), calculate the corresponding \(y\)-values, and plot the points. For example, when \(x = 0\), \(y = 5\); when \(x = 1\), \(y = 7\); and when \(x = -1\), \(y = 3\). Draw a straight line through these points.
02

Plot the second equation

Next, plot the equation \(y = -3x + 5\). Again, choose a few values for \(x\), calculate the corresponding \(y\)-values, and plot the points. For example, when \(x = 0\), \(y = 5\); when \(x = 1\), \(y = 2\); and when \(x = -1\), \(y = 8\). Draw a straight line through these points.
03

Find the intersection point

Observe the graph where the two lines intersect. The intersection point represents the solution to the system of equations, as it satisfies both equations simultaneously. Identify the coordinates of this point.
04

State the solution

Write down the coordinates of the intersection point. This point is the solution to the system of equations.

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Key Concepts

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

system of linear equations
A system of linear equations consists of two or more linear equations that share the same variables. Solving these systems means finding the values for these variables that satisfy all given equations. In our exercise, we dealt with the system:
  • \(y = 2x + 5\)
  • \(y = -3x + 5\)
intersection point
The point where the two lines cross each other on the graph is known as the intersection point. This point common to both lines provides the solution to the system of linear equations. To find the intersection point, you simply need to plot the lines and see where they meet.

In our example, the point of intersection is the solution because it satisfies both equations simultaneously. When plotted correctly, the two given lines will intersect at the point \((0, 5)\). This means that:\(x = 0\), \(y = 5\) is a common solution to both equations.
plotting coordinates
Plotting coordinates is crucial for graphing linear equations. To plot a coordinate, you need an ordered pair \((x, y)\) where 'x' indicates the position along the horizontal axis and 'y' represents the position along the vertical axis.

For each equation, follow these steps:
  • Choose a few values for \(x\) (typically, at least three points).
  • Calculate the corresponding \(y\) values using the equation.
  • Plot these \((x, y)\) points on the graph.
  • Draw a straight line through the points for each equation.
In our example, for the line \(y = 2x + 5\), we used the following points: \( (0, 5) \), \( (1, 7) \), \( (-1, 3) \). For the line \(y = -3x + 5\), we used: \( (0, 5) \), \( (1, 2) \), \( (-1, 8) \). Plotting these coordinates accurately will help you identify the intersection point visually.

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