Chapter Eight — Systems of Linear Equations and Inequalities

The Humongous Book of Algebra Problems

153

Figure 8-5: The graphs of and x – 3y = –9 both have slope and are,

therefore, parallel lines.

By deﬁnition, parallel lines do not intersect in the coordinate plane. It is

the intersection point of a system of equations that deﬁnes its solution, so the

absence of an intersection point indicates the absence of a solution to the

system. Systems of equations with no solutions are classiﬁed as “inconsistent.”

The Substitution Method

Solve one equation for a variable and plug it into the other

8.8 Explain how to apply the substitution method to solve a system of two linear

equations in two variables.

If one equation of a system can easily be solved for one of its variables, you can

substitute the resulting expression into the other equation of the system. For

instance, if the ﬁrst equation of a system can be solved for y, do so, and then

substitute the result for y in the second equation.

The net result is a new equation written in terms of a single variable, in this case

x. When solved, it produces the x-value of the solution to the system, which can

then be substituted into one of the equations of the system to determine the

corresponding y-value.

To prove

that there

is no solution

to a system of

linear equations,

all you have to

do is prove that

all the lines have

the same slope (or

that all the lines

have NO slope—

vertical lines

are parallel

as well).

You’ll learn

a couple of ways

to solve systems of

equations. This one

works best if you can

easily solve one of the

equations for x or y.

It’s even easier when

an equation is ALREADY

solved for x or y, like in

Problems 8.9 and 8.10.

Chapter Eight — Systems of Linear Equations and Inequalities

The Humongous Book of Algebra Problems

154

Note: Problems 8.9–8.10 refer to the following system of equations.

8.9 Solve the system using substitution.

The second equation in the system is solved for y: y = 11x – 16. According to this

statement, the expressions y and 11x – 16 have the same value in this system of

equations. Therefore, you may substitute 11x – 16 for y in the other equation of

the system, x + y = 8.

Solve the equation for x.

The solution to a system of linear equations in two variables is a coordinate

pair (x,y). To determine the y-coordinate, and therefore complete the solution,

substitute x = 2 into the equation y = 11x – 16.

The solution to the system of equations is (x,y) = (2,6).

Note: Problems 8.9–8.10 refer to the following system of equations.

8.10 Verify the solution generated in Problem 8.9.

According to Problem 8.9, the solution to the system of equations is

(x,y) = (2,6). To verify that the solution is correct, substitute x = 2 and y = 6

into both equations of the system.

Substituting (x,y) = (2,6) into the equations of the system produces true

statements, so it is the correct solution to the system.

You could

substitute it

into the other

equation, x + y = 8,

instead. You’ll get

the same answer.

However, it’s almost

always quickest

to plug x into the

equation already

solved for y (and

vice versa).

Chapter Eight — Systems of Linear Equations and Inequalities

The Humongous Book of Algebra Problems

155

8.11 Solve the following system of equations using substitution.

According to the ﬁrst equation, the expressions x and have the same value

in this system. Substitute for x in the second equation and solve for y.

Substitute y = 5 into the equation to complete the solution.

The solution to the system is (x,y) = (–3,5).

Rewrite the

coefcient of y (which

is 1 even though it’s not

written explicitly) using a common

denominator:

The answer

(x,y) = (5,–3) is

NOT correct. When

you write (x,y), you’re

saying that the rst

value is x and the

second value is y, and

in this problem x = –3

and y = 5, not

the other way

around.

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