5.2 Solve systems of equations by substitution (Page 2/5)

Elementary algebra Page 2 / 5

Solve the system by substitution. ${\begin{matrix} 3 x + y = 5 \\ 2 x + 4 y = −10 \end{matrix}$

Solution

We need to solve one equation for one variable. Then we will substitute that expression into the other equation.

Solve for y . Substitute into the other equation.
Replace the y with −3 x + 5.
Solve the resulting equation for x .

Substitute x = 3 into 3 x + y = 5 to find y .

The ordered pair is (3, −4).
Check the ordered pair in both equations: $\begin{matrix} \begin{matrix} 3 x + y & = & 5 \\ 3 \cdot 3 + (- 4) & \overset{?}{=} & 5 \\ 9 - 4 & \overset{?}{=} & 5 \\ 5 & = & 5 ✓ \end{matrix} & \begin{matrix} 2 x + 4 y & = & - 10 \\ 2 \cdot 3 + 4 (- 4) & = & - 10 \\ 6 - 16 & \overset{?}{=} & - 10 \\ - 10 & = & - 10 ✓ \end{matrix} \end{matrix}$
	The solution is (3, −4).

Got questions? Get instant answers now!

Solve the system by substitution. ${\begin{matrix} 4 x + y = 2 \\ 3 x + 2 y = −1 \end{matrix}$

$(1, −2)$

Got questions? Get instant answers now!

Solve the system by substitution. ${\begin{matrix} - x + y = 4 \\ 4 x - y = 2 \end{matrix}$

$(2, 6)$

Got questions? Get instant answers now!

In [link] it was easiest to solve for y in the first equation because it had a coefficient of 1. In [link] it will be easier to solve for x .

Solve the system by substitution. ${\begin{matrix} x - 2 y = −2 \\ 3 x + 2 y = 34 \end{matrix}$

Solution

We will solve the first equation for $x$ and then substitute the expression into the second equation.


Solve for x . Substitute into the other equation.
Replace the x with 2 y − 2.
Solve the resulting equation for y .
Substitute y = 5 into x − 2 y = −2 to find x .
The ordered pair is (8, 5).
Check the ordered pair in both equations: $\begin{matrix} \begin{matrix} x - 2 y & = & - 2 \\ 8 - 2 \cdot 5 & \overset{?}{=} & - 2 \\ 8 - 10 & \overset{?}{=} & - 2 \\ - 2 & = & - 2 ✓ \end{matrix} & \begin{matrix} 3 x + 2 y & = & 34 \\ 3 \cdot 8 + 2 \cdot 5 & \overset{?}{=} & 34 \\ 24 + 10 & \overset{?}{=} & 34 \\ 34 & = & 34 ✓ \end{matrix} \end{matrix}$
	The solution is (8, 5).

Got questions? Get instant answers now!

Solve the system by substitution. ${\begin{matrix} x - 5 y = 13 \\ 4 x - 3 y = 1 \end{matrix}$

$(−2, −3)$

Got questions? Get instant answers now!

Solve the system by substitution. ${\begin{matrix} x - 6 y = −6 \\ 2 x - 4 y = 4 \end{matrix}$

$(6, 2)$

Got questions? Get instant answers now!

When both equations are already solved for the same variable, it is easy to substitute!

Solve the system by substitution. ${\begin{matrix} y = −2 x + 5 \\ y = \frac{1}{2} x \end{matrix}$

Solution

Since both equations are solved for y , we can substitute one into the other.

Substitute $\frac{1}{2} x$ for y in the first equation.
Replace the y with $\frac{1}{2} x .$
Solve the resulting equation. Start by clearing the fraction.
Solve for x .

Substitute x = 2 into y = $\frac{1}{2} x$ to find y .
The ordered pair is (2,1).
Check the ordered pair in both equations: $\begin{matrix} \begin{matrix} y & = & \frac{1}{2} x \\ 1 & \overset{?}{=} & \frac{1}{2} \cdot 2 \\ 1 & = & 1 ✓ \end{matrix} & \begin{matrix} y & = & - 2 x + 5 \\ 1 & \overset{?}{=} & - 2 \cdot 2 + 5 \\ 1 & = & - 4 + 5 \\ 1 & = & 1 ✓ \end{matrix} \end{matrix}$
	The solution is (2,1).

Got questions? Get instant answers now!

Solve the system by substitution. ${\begin{matrix} y = 3 x - 16 \\ y = \frac{1}{3} x \end{matrix}$

$(6, 2)$

Got questions? Get instant answers now!

Solve the system by substitution. ${\begin{matrix} y = − x + 10 \\ y = \frac{1}{4} x \end{matrix}$

$(8, 2)$

Got questions? Get instant answers now!

Be very careful with the signs in the next example.

Solve the system by substitution. ${\begin{matrix} 4 x + 2 y = 4 \\ 6 x - y = 8 \end{matrix}$

Solution

We need to solve one equation for one variable. We will solve the first equation for y .


Solve the first equation for y .
Substitute −2 x + 2 for y in the second equation.
Replace the y with −2 x + 2.
Solve the equation for x .

Substitute $x = \frac{5}{4}$ into 4 x + 2 y = 4 to find y .
The ordered pair is $(\frac{5}{4}, - \frac{1}{2}) .$
Check the ordered pair in both equations. $\begin{matrix} \begin{matrix} 4 x + 2 y & = & 4 \\ 4 (\frac{5}{4}) + 2 (- \frac{1}{2}) & \overset{?}{=} & 4 \\ 5 - 1 & \overset{?}{=} & 4 \\ 4 & = & 4 ✓ \end{matrix} & \begin{matrix} 6 x - y & = & 8 \\ 6 (\frac{5}{4}) - (- \frac{1}{2}) & \overset{?}{=} & 8 \\ \frac{15}{4} - (- \frac{1}{2}) & \overset{?}{=} & 8 \\ \frac{16}{2} & \overset{?}{=} & 8 \\ 8 & = & 8 ✓ \end{matrix} \end{matrix}$
	The solution is $(\frac{5}{4}, - \frac{1}{2}) .$

Got questions? Get instant answers now!

Solve the system by substitution. ${\begin{matrix} x - 4 y = −4 \\ −3 x + 4 y = 0 \end{matrix}$

$(2, \frac{3}{2})$

Got questions? Get instant answers now!

Solve the system by substitution. ${\begin{matrix} 4 x - y = 0 \\ 2 x - 3 y = 5 \end{matrix}$

$(- \frac{1}{2}, −2)$

Got questions? Get instant answers now!

In [link] , it will take a little more work to solve one equation for x or y .

Solve the system by substitution. ${\begin{matrix} 4 x - 3 y = 6 \\ 15 y - 20 x = −30 \end{matrix}$

Solution

We need to solve one equation for one variable. We will solve the first equation for x .


Solve the first equation for x .
Substitute $\frac{3}{4} y + \frac{3}{2}$ for x in the second equation.
Replace the x with $\frac{3}{4} y + \frac{3}{2} .$
Solve for y .

Since 0 = 0 is a true statement, the system is consistent. The equations are dependent. The graphs of these two equations would give the same line. The system has infinitely many solutions.

Got questions? Get instant answers now!

Questions & Answers

Explain the following terms . (1) Abiotic factors in an ecosystem

Nomai Reply

Abiotic factors are non living components of ecosystem.These include physical and chemical elements like temperature,light,water,soil,air quality and oxygen etc

Qasim

what is biology

daniel Reply

what is diffusion

Emmanuel Reply

passive process of transport of low-molecular weight material according to its concentration gradient

AI-Robot

what is production?

Catherine

Pathogens and diseases

how did the oxygen help a human being

Achol Reply

how did the nutrition help the plants

Achol Reply

Biology is a branch of Natural science which deals/About living Organism.

Ahmedin Reply

what is phylogeny

Odigie Reply

evolutionary history and relationship of an organism or group of organisms

AI-Robot

Deng

what is biology

Hajah Reply

cell is the smallest unit of the humanity biologically

Abraham

what is biology

Victoria Reply

what is biology

Abraham

HOW CAN MAN ORGAN FUNCTION

Alfred Reply

the diagram of the digestive system

Assiatu Reply

allimentary cannel

Ogenrwot

How does twins formed

William Reply

They formed in two ways first when one sperm and one egg are splited by mitosis or two sperm and two eggs join together

Oluwatobi

what is respiration

Deborah

what is genetics

Josephine Reply

Genetics is the study of heredity

Misack

how does twins formed?

Misack

What is manual

Hassan Reply

Got questions? Join the online conversation and get instant answers!

Jobilize.com Reply

<< Chapter < Page Page > Chapter >>

Read also:

Get Jobilize Job Search Mobile App in your pocket Now!

100% Free Mobile Applications
Receive real-time job alerts and never miss the right job again

Source: OpenStax, Elementary algebra. OpenStax CNX. Jan 18, 2017 Download for free at http://cnx.org/content/col12116/1.2

Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Elementary algebra' conversation and receive update notifications?

Ask

	38 Biology 38 The Musculoskeletal System MCQ By OpenStax Start Quiz
	13 Lec:13 Hypothesis Testing P-values By Janet Forrester Start Quiz
	Anthropology Economic System By Richley Crapo Start Assignment
	Negotiations Conflict Management BUS403 By Charles Jumper Start Quiz
	27 AP 27 Reproductive System Essay By OpenStax Start Flashcards
	1 AP 01 Human Body Anatomy Physiology MCQ By OpenStax Start Quiz
	47 Biology 47 Conservation Biology Biodiversity MCQ By OpenStax Start Quiz
	2 CDL Quiz - Air Brakes Endorsement Part 2 By Jazzycazz Jackson Start Quiz
	2 BOD - Neuropathology By Brooke Delaney Start Exam
	9 AP 09 Joints Essay Quiz By OpenStax Start Flashcards