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Chapter 4: Problem 34

Solve by graphing. $$ \begin{aligned} &y=3 x+4 \\ &y=-2 x+4 \end{aligned} $$

Short Answer

Expert verified
The solution is \((0, 4)\).

Step by step solution

01

Understand the Problem

We need to find the solution to the system of linear equations by graphing the following equations: 1. \( y = 3x + 4 \)2. \( y = -2x + 4 \)
02

Graph the First Equation

Graph the equation \( y = 3x + 4 \) by finding points that satisfy the equation. For example, when \( x = 0 \), \( y = 4 \). Another point would be when \( x = 1 \), \( y = 7 \). Plot these points and draw the line.
03

Graph the Second Equation

Graph the equation \( y = -2x + 4 \) by finding points that satisfy the equation. For example, when \( x = 0 \), \( y = 4 \). Another point would be when \( x = 1 \), \( y = 2 \). Plot these points and draw the line.
04

Find the Intersection

The solution to the system of equations is the point where the two lines intersect. From the graphs of \( y = 3x + 4 \) and \( y = -2x + 4 \), we see that they intersect at the point \((0, 4)\).

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Key Concepts

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

solving systems of equations
When solving systems of equations, you are finding the values of variables that satisfy all equations in a system simultaneously. A common method for solving systems of equations is graphing. This involves plotting each equation on a graph to find the point(s) where they intersect.

For the given system:
1. \( y = 3x + 4 \)
2. \( y = -2x + 4 \)
we need to find a common \( x \) and \( y \) that make both equations true.

We can solve by graphing these two linear equations on a coordinate plane and looking for their point of intersection. This point is the solution to our system.
intersection of lines
The intersection of lines refers to the point where two lines cross each other on a graph. For our system, we want to find where the lines described by the equations \( y = 3x + 4 \) and \( y = -2x + 4 \) intersect.

Steps to find the intersection:
- Plot the first line (\

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