Problem 27 Solve a System of Linear Equatio... [FREE SOLUTION] (2024)

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Chapter 5: Problem 27

Solve a System of Linear Equations by Graphing In the following exercises,solve the following systems of equations by graphing. $$ \left\\{\begin{array}{l} x+y=-4 \\ -x+2 y=-2 \end{array}\right. $$

Short Answer

Expert verified

The solution is \( (-2, -2) \).

Step by step solution

01

Write the equations

Given the system of equations: \( x + y = -4 \) \( -x + 2y = -2 \)

03

Convert the second equation to slope-intercept form

\( -x + 2y = -2 \) Add \( x \) to both sides: \( 2y = x - 2 \) Divide both sides by 2: \( y = \frac{1}{2}x - 1 \)

04

Graph the first equation

Graph the equation \( y = -x - 4 \). Start at the y-intercept \( (0, -4) \). Use the slope to plot another point: From \( (0, -4) \), go down 1 unit and right 1 unit to point \( (1, -5) \).

05

Graph the second equation

Graph the equation \( y = \frac{1}{2}x - 1 \). Start at the y-intercept \( (0, -1) \). Use the slope to plot another point: From \( (0, -1) \), go up 1 unit and right 2 units to point \( (2, 0) \).

06

Find the intersection of the lines

The intersection of the two graphs represents the solution to the system of equations. The lines intersect at the point \( (-2, -2) \).

07

Verify the solution

Substitute \( x = -2 \) and \( y = -2 \) into both original equations to verify: For \( x + y = -4 \): \( -2 + (-2) = -4 \) which is true. For \( -x + 2y = -2 \): \( -(-2) + 2(-2) = -2 \) which is true.

Key Concepts

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

Linear Equations

A linear equation is a type of equation that creates a straight line when graphed on a coordinate plane. It has the general form of \(Ax + By = C\), where \(A\), \(B\), and \(C\) are constants. In a system of linear equations, you have two or more linear equations that you deal with at the same time. For example, in the given exercise, the system consists of two linear equations:

  • \(x + y = -4\)
  • \(-x + 2y = -2\)

To solve a system of linear equations by graphing, you graph each equation on the same set of axes and look for the point where the lines intersect.

Graphing Lines

When you graph a linear equation, you convert it into a format that easily shows its characteristics. The most common form for this is the slope-intercept form, \(y = mx + b\), where \(m\) represents the slope and \(b\) represents the y-intercept. For the given exercise, we converted both equations:

  • \(x + y = -4\) becomes \(y = -x - 4\)
  • \(-x + 2y = -2\) becomes \(y = \frac{1}{2}x - 1\)

Once they are in slope-intercept form, you can easily plot the y-intercepts and use the slopes to find other points on each line.

Intersection of Lines

The intersection of two lines is the point where they cross each other. This point represents the solution to the system of equations because it is the only point that satisfies both equations simultaneously. In the exercise, after graphing the equations \(y = -x - 4\) and \(y = \frac{1}{2}x - 1\), we found they intersect at \((-2, -2)\). This implies that \(x = -2\) and \(y = -2\) is the solution to the system.

Y-Intercept

The y-intercept of a line is the point where the line crosses the y-axis, which means the value of \(x\) at this point is zero. For example, in the equation \(y = -x - 4\), the y-intercept is \( -4 \), so the line crosses the y-axis at (0, -4). In the equation \(y = \frac{1}{2}x - 1\), the y-intercept is \( -1 \), so the line crosses the y-axis at (0, -1). Starting with the y-intercept makes it easier to draw the line on the coordinate plane.

Slope

The slope of a line describes how steep the line is and the direction it goes (up or down). It is calculated as the change in \(y\) over the change in \(x\) (\(\frac{ \text{rise} }{ \text{run} }\)). In the slope-intercept form \(y = mx + b\), \(m\) represents the slope. A positive slope means the line inclines upwards, while a negative slope means it declines downward. In the exercise:

  • For \(y = -x - 4\), the slope is \( -1 \).
  • For \(y = \frac{1}{2}x - 1\), the slope is \( \frac{1}{2} \).

Using the slope, you can plot additional points starting from the y-intercept to ensure accuracy when graphing the line.

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Problem 27 Solve a System of Linear Equatio... [FREE SOLUTION] (3)

Most popular questions from this chapter

In the following exercises, translate to a system of equations and solve. The difference of two complementary angles is 17 degrees. Find the measures ofthe angles.In the following exercises, translate to a system of equations and solve. Two angles are supplementary. The measure of the larger angle is four morethan three times the measure of the smaller angle. Find the measures of bothangles.In the following exercises, translate to a system of equations and solve. A cashier has 30 bills, all of which are \(\$ 10\) or \(\$ 20\) bills. The totalvalue of the money is \(\$ 460\). How many of each type of bill does the cashierhave?After four years in college, Josie owes \(\$ 65,800\) in student loans. Theinterest rate on the federal loans is \(4.5 \%\) and the rate on the privatebank loans is \(2 \%\). The total interest she owed for one year was \(\$2,878.50 .\) What is the amount of each loan?
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Problem 27 Solve a System of Linear Equatio... [FREE SOLUTION] (2024)

FAQs

How do you solve a system of linear equations without solution? ›

System of Linear Equations with No Solutions

The red line is the graph of y=4x+2, and the blue line is the graph of y=4x+5. When two equations have the same slope but different y-axis, they are parallel. Since there are no intersection points, the system has no solutions.

What is the solution to this system of linear equations 3x 2y 14 5x y 32? ›

Summary: The solution to this system of linear equations 3x – 2y = 14 and 5x + y = 32 is (x, y) is (6, 2).

What is the solution to this system of linear equations 2x 3y 3 7x 3y 24? ›

The solution to this system of linear equations 2x+3y=3 and 7x-3y=24 is (3, -1).

What is the solution to this system of linear equations y − 4x 7 2y 4x 2? ›

Expert-Verified Answer

So, the solution to the linear equation y – 4x = 7 and 2y + 4x = 2 is (-1, 3). To solve the linear equations in two variables is by adding both the equation. Then, substitute the value of y in equation (1).

What is the easiest way to solve a linear system? ›

SOLVE A SYSTEM OF LINEAR EQUATIONS BY GRAPHING.
  1. Graph the first equation.
  2. Graph the second equation on the same rectangular coordinate system.
  3. Determine whether the lines intersect, are parallel, or are the same line.
  4. Identify the solution to the system. ...
  5. Check the solution in both equations.
Nov 24, 2022

What is the formula for no solution? ›

Condition for No Solution:

Considering the pair of linear equations by two variables u and v. Therefore a1, b1, c1, a2, b2, c2 are real numbers. If (a1/a2) = (b1/b2) ≠ (c1/c2), then this will result in no solution.

How many solutions does this linear system have 6x 2 12x 2y 4? ›

The linear system of equations y = -6x + 2 and -12x - 2y = -4 has infinitely many solutions.

What is the solution to the system of equations answer? ›

The solution set to a system of equations will be the coordinates of the ordered pair(s) that satisfy all equations in the system. In other words, those values of x and y will make the equations true. Accordingly, when a system of equations is graphed, the solution will be all points of intersection of the graphs.

How do you find the solution to a linear equation? ›

The steps for solving linear equations are:
  1. Simplify both sides of the equation and combine all same-side like terms.
  2. Combine opposite-side like terms to obtain the variable term on one side of the equal sign and the constant term on the other.
  3. Divide or multiply as needed to isolate the variable.
  4. Check the answer.
Oct 6, 2021

What is solve the system of linear equations? ›

A solution of a linear system is an assignment of values to the variables x1, x2, ..., xn such that each of the equations is satisfied. The set of all possible solutions is called the solution set. A linear system may behave in any one of three possible ways: The system has infinitely many solutions.

What is the solution of this linear system? ›

A system of linear equations consists of the equations of two lines. The solution to a system of linear equations is the point which lies on both lines. In other words, the solution is the point where the two lines intersect.

What is a system of linear equations that has no solution called? ›

A system of equations that has no solution is called an inconsistent system. This means that there is no set of values that can be assigned to the variables in the equations that would make all of the equations in the system true at the same time.

How to know if a linear system has no solution without graphing? ›

Two equations have parallel lines (no solution to the system) if the slopes are equal and and y-intercepts are not. Adding the equations gives an obviously false statement. This system of equations has no solution.

How do you solve a system of linear equations without graphing? ›

To solve a system of linear equations without graphing, you can use the substitution method. This method works by solving one of the linear equations for one of the variables, then substituting this value for the same variable in the other linear equation and solving for the other variable.

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