Envision Math Grade 8 Volume 1 Chapter 5 Analyze And Solve System Of Linear Equations
Page 287 Essential Question Answer
Find: to solve a system of linear equation.
A system of linear equations is formed by two or more linear equations that use the same variables.
A linear system solution is the assignment of values to variables in such a way that all the equations are concurrently fulfilled.
Any ordered pair that makes all equations in the system true is a solution of a system of linear equations.
A system of linear equations is formed by two or more linear equations that use the same variable.
Take, for example, a system of linear equations:
\(\left\{\begin{array}{l}
x+y=3 \\
x-y=1
\end{array}\right.\)
The solution of the system is (2,1).
Any ordered pair that makes all equations in the system true simultaneously is a solution of a system of linear equations.
A system of linear equations is formed by two or more linear equations that use the same variables.
Page 287 Use Vocabulary In Writing Answer
Find: Use vocabulary terms to find the number of solutions of two or more equations by using the slope and the y-intercept.
The lines will be parallel if the two linear equations have the same slope but distinct y-intercepts.
Because parallel lines never overlap, a system made up of two parallel lines has no solution.
If two linear equations have the same slope and y-intercept, they describe the same line.
There are an unlimited number of solutions since a line crosses itself everywhere.
In any other scenario when the slope varies, the system of equations will have a single solution.
There will be no solution for linear equations with the same slope but a different y-intercept.
There are an unlimited number of solutions to linear equations with the same slope and y-intercept.
In any other scenario, when the slope varies, the system of equations will have a single solution.
Page 288 Exercise 2 Answer
Given:
y = 2x + 10
3y − 6x = 30
Find: equations has one solution, no solution, or infinitely many solutions.
The relationship between the lines and the number of solutions are determined by the slopes and y-intercepts of the linear equations in a system.
The equation y = 2x + 10 is written in slope-intercept form.
Since both equations are the same, they intersect at every point, so the system has infinitely many solutions.
Since both equations are the same, they intersect at every point, so the system has infinitely many solutions.
Page 288 Exercise 3 Answer
Given
\(-3 x+\frac{1}{3} y=12\)
2y = 18x + 72
Find: equations has one solution, no solution, or infinitely many solutions.
The relationship between the lines and the number of solutions are determined by the slopes and y-intercepts of the linear equations in a system.
Since both equations are the same, they intersect at every point, so the system has infinitely many solutions.
Since both equations are the same, they intersect at every point, so the system has infinitely many solutions.
Page 288 Exercise 4 Answer
Given:
\(y-\frac{1}{4} x=-1\)
y – 2 = 4x
Find: equations has one solution, no solution, or infinitely many solutions.
The relationship between the lines and the number of solutions are determined by the slopes and y-intercepts of the linear equations in a system.
Rewrite the first equation in slope-intercept form,
The equations of the linear system have different slopes.
Therefore, the system has no solution.
Since both equations are the not same, they don’t intersect at every point, so the system has no solutions.
Page 288 Exercise 5 Answer
Given:
Turkey costs $3 per pound at Store A and $4.50 per pound at Store B.
Ham costs $4 per pound at Store B and $6 per pound at Store B.
Michael spends $18 at Store A, and Ashley spends $27 at Store B.
Find: equations has one solution, no solution, or infinitely many solutions.
The relationship between the lines and the number of solutions are determined by the slopes and y-
Let x be the amount of the first kind of meat and y be the second kind of the meat.
Since the boy pays $18 in the first shop, the first equation of the system representing the situation,
3x + 4y = 18
Since the girl pays $27 in the second shop, the second equation of the system representing the situation,
4.5x + 6y = 27
The system of equations,
Since both equations are the same, they intersect at every point, so the system has infinitely many solutions.
Since the system has infinitely many solutions, it follows that there are infinitely many ways for them to buy the same amount of both kinds of meat.
Page 289 Exercise 1 Answer
Given:
y = \(\frac{1}{2} x+1\)
-2x + 4y = 4
Find: graph each system and find the solution.
The relationship between the lines and the number of solutions are determined by the slopes and y-intercepts of the linear equations in a system.
Write the second equation in slope-intercept form,
Using the same method, calculate the values of y for a few different values of x,
Plot the points (−4,−1),(−2,0),(0,1),(2,2) and (4,3) as shown in the below graph:
Each point on the line represents a solution in the above graph.
Since both lines overlap, the system has infinitely many solutions.
The system has infinitely many solutions.
The required graph is:
Page 289 Exercise 2 Answer
Given:
y = −x − 3
y + x = 2
Find: graph each system and find the solution.
The relationship between the lines and the number of solutions are determined by the slopes and y-intercepts of the linear equations in a system.
The first equation in the slope-intercept form,
y = −x − 3
Calculate the value of y for x = 0,
y = −x − 3
y = −0 − 3
y = −3
Using the same method, calculate the values of y for a few other values of x,
Plot the points (0,−3),(1,−4),(2,−5) and (3,−6) and joins the points as shown in the below graph:
The above graph show the slope intercept of y = −x − 3.
Write the second equation in slope-intercept form,
y + x = 2
y = 2 − x
y = −x + 2
Graph the line y + x = 2 as shown in the below graph:
The above graph show the slope intercept of y = −x − 3 and y = −x + 2.
Lines y = −x + 2 and y = −x − 3 are parallel, so the system has no solution.
The required graph:
Page 289 Exercise 3 Answer
Given:
2y = 6x + 4
y = −2x + 2
Find: graph each system and find the solution.
The relationship between the lines and the number of solutions are determined by the slopes and y-intercepts of the linear equations in a system.
To graph the first equation 2y = 6x + 4, find the intercepts of the graph of the equation,
2y = 6x + 4
Substitute x = 0 into the equation to find the y-intercept,
2y = 6⋅0 + 4
2y = 4
y = 2
Since the y-intercept is 2, the first point on the graph of the first equation is (0,2).
Now substitute the value of y = 0 in 2y = 6x + 4,
2(0) = 6x + 4
2(0) – 4 = 6x
-4 = 6x
\(-\frac{2}{3} = x-1\)
Since the x-intercept \(-\frac{2}{3}\), the second point on the graph of the first equation is \(\left(-\frac{2}{3}, 0\right)\).
Plot the points (0, 2) and \left(-\frac{2}{3}, 0\right) on the graph as shown below:
To graph the second equation y = −2x + 2, find the intercepts of the graph of the equation,
y = −2x + 2
Substitute x = 0 into the equation to find the y-intercept,
y = −2⋅0 + 2
y = −0 + 2
y = 2
Since the y-intercept is 2, the first point on the graph of the second equation is (0,2).
Substitute y = 0 in the equation y = −2x + 2,
y = −2x + 2
0 = −2x + 2
2x = 2
x = 1
Since the x-intercept is 1, the second point on the graph of the second equation is (1,0).
Plot the points(1,0) and (0,2) on the graph as shown below:
The above graph show the slope intercept of y = −2x + 2.
The graph of 2y = 6x + 4 and y = −2x + 2 are shown below:
The above graph shows that the intersection point is (0,2).
Substitute x = 0 and y = 2 into the first equation 2y = 6x + 4,
2(2) = 6(0) + 4
4 = 0 + 4
4 = 4
It means that the statement is true.
Substitute x = 0 and y = 2 into the first equation y = −2x + 2,
y = −2x + 2
2 = −2(0) + 2
2 = 0 + 2
2 = 2
It means that the statement is true.
Since both statements are true, the point (0,2) is the solution to the system.
The solution of the given system of equation is (0,2).
The required graph:
Page 290 Exercise 1 Answer
Given:
−3y = −2x − 1
y = x − 1
Find: solve each system.
The solution of the given equation is (x, y) = (4, 3).
Page 290 Exercise 2 Answer
Given:
y = 5x + 2
2y − 4 = 10x
Find: solve each system.
It means the equation has infinite, many solutions.
The solution of the given equation has infinitely many solutions.
Page 290 Exercise 3 Answer
Given:
2y − 8 = 6x
y = 3x + 2
Find: solve each system.
This means the equation has no solution.
There is no solution for the given system of equation. That is x ∈ ∅.
Page 290 Exercise 4 Answer
Given:
2y − 2 = 4x
y = −x + 4
Find: solve each system.
The solution of the given equation is (x, y)=(1,3).
Page 290 Exercise 1 Answer
Given:
−2x + 2y = 2
4x − 4y = 4
Find: solve the equation.
There is no solution for the equation.
The statement is false for any value of x and y, so there is no solution. That is (x, y)∈ ∅.
Page 290 Exercise 2 Answer
Given:
4x + 6y = 40
−2x + y = 4
Find: solve the equation.
The solution of the given equation is (x, y) = (1, 6).
Page 290 Exercise 3 Answer
Given:
A customer at a concession stand bought 2 boxes of popcorn and 3 drinks for $12.
Another customer bought 3 boxes of popcorn and 5 drinks for $19.
Find:
Box of popcorn cost? Drink cost?
Use the elimination method to solve the equation.
Let x be the number of box of popcorn and y the number of drinks.
Since the cost of 2 box of popcorn plus the cost of 3 drinks is $12,
2x + 3y = 12.
And since the cost of 3 box of popcorn plus the cost of 5 drinks is $19,3x + 5y = 19.
A system of equations representing the situation,
The box of popcorn cost $3.
The drink cost $2.