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Chapter 5: Problem 3

Solve each system of equations by sketching the graphs. Use the slope and the \(y\) -intercept or both intercepts. Estimate each result to the nearest 0,1 if necessary. $$\begin{aligned} &y=-x+4\\\ &y=x-2 \end{aligned}$$

Short Answer

Expert verified
The lines intersect at (3, 1).

Step by step solution

01

Understanding the Equations

The given system of equations consists of two linear equations: (1) \( y = -x + 4 \) and (2) \( y = x - 2 \). These equations represent straight lines when graphed on a coordinate plane. Our task is to find the point where these two lines intersect.
02

Identify Slopes and Intercepts

For equation (1) \( y = -x + 4 \), the slope is \(-1\) and the \( y \)-intercept is \( 4 \). For equation (2) \( y = x - 2 \), the slope is \( 1 \) and the \( y \)-intercept is \(-2\). These values will help us graph the lines accurately.
03

Sketch the First Equation

Start by plotting the \( y \)-intercept (0, 4) for the first line, \( y = -x + 4 \). Since the slope is \(-1\), we move one unit down and one unit to the right to find another point such as (1, 3). Draw the line through these points.
04

Sketch the Second Equation

Now plot the \( y \)-intercept (0, -2) for the second line, \( y = x - 2 \). With a slope of \( 1 \), move one unit up and one unit to the right to find another point like (1, -1). Draw the line through these points.
05

Identify the Intersection Point

Find the point where the two lines cross. From the graph, this intersection occurs at (3, 1). This is where both equations are satisfied with the same \( x \) and \( y \) values.

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Key Concepts

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

Linear Equations
A linear equation represents a straight line on a graph. It's written in the form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. Linear equations are fundamental in algebra and describe relationships where one variable changes at a constant rate with respect to another. In a system of linear equations, you have more than one linear equation. The goal is often to find a point that satisfies all equations simultaneously, which means finding where their graphs intersect on a coordinate plane.
Graphing Solutions
Graphing solutions is a visual method to solve systems of equations, especially linear ones. It involves plotting each equation on the same coordinate plane and identifying where the graphs intersect. Graphing gives a clear picture of how two or more lines (or curves) relate to each other.

To graph an equation: - Start with the y-intercept, which is the point where the line crosses the y-axis. - Use the slope to find another point. - Draw a line through these points. Graphing provides an intuitive understanding and can be particularly helpful when you need to estimate solutions, even if they're not exact.
Slopes and Intercepts
The slope of a line describes its steepness and direction. For instance, a slope of -1 means the line descends diagonally, moving one unit down for each unit it moves right. A slope of 1 ascends diagonally, moving up one unit for every unit to the right.

Slopes can tell us: - If the lines will intersect (if they have different slopes). - If the lines are parallel (same slope but different intercepts). - If the lines are identical (same slope and intercepts).

The y-intercept is where the line crosses the y-axis. It's a crucial point to start plotting the linear equation. Together, slopes and intercepts help efficiently graph lines and find intersection points.
Coordinates of Intersection
The coordinates of intersection play a key role in solving systems of linear equations. This point is where the graphs of the equations meet, and it satisfies all equations in the system simultaneously.

For example, to find where two lines intersect, graph each line and see where they cross. The cross-point's coordinates are the solution to the system. In our case, the intersection of the lines from the equations \( y = -x + 4 \) and \( y = x - 2 \) is at (3, 1).

These coordinates describe a real-world relationship or situation where both conditions/expressions hold true. Understanding how to find these coordinates helps in solving more complex problems using linear systems.

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Most popular questions from this chapter

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