Problem 8 For the linear system \(\left\\{... [FREE SOLUTION]

Chapter 3: Problem 8

For the linear system \(\left\\{\begin{array}{l}x-y=5 \\ 2 x+y=1\end{array}\right.\) a. Graph the system. Estimate the solution for the system and then find the exact solution. b. Check that your solution satisfies both of the original equations.

Short Answer

Expert verified

The exact solution is (2, -3), and it satisfies both original equations.

Step by step solution

Write the Equations in Slope-Intercept Form

Rewrite each equation of the system in the form of y = mx + b.Equation 1: x - y = 5Rearrange to get y: y = x - 5Equation 2: 2x + y = 1Rearrange to get y:y = -2x + 1

Graph Each Equation on the Same Coordinate Plane

Plot the points for each equation.For y = x - 5, use points (0, -5) and (5, 0).For y = -2x + 1, use points (0, 1) and (1, -1).

Estimate the Solution from the Graph

Identify the point where the two lines intersect on the graph. This is the estimated solution for the system of equations.

Solve the System Algebraically

Use the substitution or elimination method to find the exact solution.Using substitution: From Equation 1, y = x - 5.Substitute y in Equation 2:2x + (x - 5) = 1Combine like terms:3x - 5 = 1Solve for x:3x = 6x = 2Substitute x = 2 into Equation 1 to find y:y = 2 - 5y = -3The exact solution is the point (2, -3).

Check the Solution

Verify that (2, -3) satisfies both original equations.Equation 1: x - y = 52 - (-3) = 55 = 5 (True)Equation 2: 2x + y = 12(2) + (-3) = 14 - 3 = 11 = 1 (True)Therefore, the solution (2, -3) satisfies both equations.

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Key Concepts

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

graphing linear equations

Graphing linear equations is a visual way to solve systems of equations. It allows you to see where the lines representing each equation intersect. To graph a linear equation, you need to find at least two points that satisfy the equation.

For example, let's graph the equation y = x - 5. We can choose x-values and solve for y to get points:

If x = 0, then y = -5, giving us the point (0, -5).
If x = 5, then y = 0, giving us the point (5, 0).

Similarly, for the equation y = -2x + 1:

If x = 0, then y = 1, giving us the point (0, 1).
If x = 1, then y = -1, giving us the point (1, -1).

Plot these points on a graph and draw a line through them. The point where the two lines intersect represents the solution to the system of equations.

slope-intercept form

The slope-intercept form is a way of writing linear equations as y = mx + b. Here, m represents the slope and b represents the y-intercept.

The slope (m) describes how steep the line is. It's the change in y divided by the change in x.
The y-intercept (b) is where the line crosses the y-axis. This form is useful because it quickly lets you see the slope and y-intercept.

For example, for the equation y = x - 5, the slope is 1 (since it's 1*x) and the y-intercept is -5. For y = -2x + 1, the slope is -2 and the y-intercept is 1.

This information helps you graph the equations because you know the starting point (y-intercept) and how the line moves (slope).

substitution method

The substitution method solves a system of equations by substituting one equation into the other. Here's how:

Step 1: Solve one of the equations for one variable.
Step 2: Substitute this expression into the other equation.
Step 3: Solve the resulting equation for the remaining variable.
Step 4: Substitute back to find the other variable.

Let's use our system:

From equation 1: x - y = 5, solve for y: y = x - 5.
Substitute this into equation 2: 2x + y = 1.
2x + (x - 5) = 1, combine like terms: 3x - 5 = 1.
Add 5 to both sides: 3x = 6.
Solve for x: x = 2.

Now substitute x = 2 back into y = x - 5:

y = 2 - 5 = -3.

So, the solution is (2, -3).

elimination method

The elimination method solves systems by adding or subtracting equations to eliminate one variable. Here's a step-by-step process:

Step 1: Arrange the equations with like terms in columns.
Step 2: Multiply one or both equations to get coefficients of a variable to be the same.
Step 3: Add or subtract the equations to eliminate one variable.
Step 4: Solve for the remaining variable, then back-substitute to find the other.

For our system:

Equation 1: x - y = 5
Equation 2: 2x + y = 1

Add the two equations to eliminate y:

(x - y) + (2x + y) = 5 + 1.
3x = 6, solve for x: x = 2.

Substitute x = 2 back into Equation 1:

x - y = 5.
2 - y = 5, solve for y: y = -3.

Thus, the solution is (2, -3).

91影视

For the linear system \(\left\\{\begin{array}{l}x-y=5 \\ 2 x+y=1\end{array}\right.\) a. Graph the system. Estimate the solution for the system and then find the exact solution. b. Check that your solution satisfies both of the original equations.

Short Answer

Step by step solution

Write the Equations in Slope-Intercept Form

Graph Each Equation on the Same Coordinate Plane

Estimate the Solution from the Graph

Solve the System Algebraically

Check the Solution

Key Concepts

graphing linear equations

slope-intercept form

substitution method

elimination method

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