# SOLVING SYSTEMS BY SUBSTITUTION

The method of substitution can be used to solve systems of linear equations by solving an equation for one variable and then substituting the resulting expression for that variable into the other equation.

The following steps will be useful to solve system of equations by substitution.

Step 1 :

Solve one of the equations for one of its variables.

Step 2 :

Substitute the expression from step 1 into the other equation and solve for the other variable.

Step 3 :

Substitute the value from step 2 into either original equation and solve to find the value of the variable in step 1.

## Solved Examples

Example 1 :

Solve the system of linear equations by substitution. Check your answer by graphing.

4x + y  =  8

-3x + y  =  1

Solution :

Step 1 :

Solve an equation for one variable.

Select one of the equation, say -3x + y  =  1.

Solve for the variable y in terms of x.

(-3x + y) + 3x  =  (1) + 3x

-3x + y + 3x  =  1 + 3x

Simplify.

y  =  1 + 3x

Step 2 :

Substitute the expression for y in the other equation and solve.

4x + y  =  8

4x + (1 + 3x)  =  8

Combine like terms.

7x + 1  =  8

Subtract 1 from both sides.

7x  =  7

Divide both sides by 7.

7x / 7  =  7 / 7

x  =  1

Step 3 :

Substitute the value of x we got above (x = 1) into one of the equations and solve for the other variable, y.

4x + y  =  8

4(1) + y  =  8

4 + y  =  8

Subtract 4 from both sides.

y  =  8

Hence, the solution of the system is (1, 8).

Step 4 :

Check the solution by graphing.

To graph the equations, write them in slope-intercept form.

That is,

y  =  mx + b

4x + y  =  8

y  =  - 4x + 8

Slope  =  - 4

y-intercept  =  8

-3x + y  =  1

y  =  3x + 1

Slope  =  3

Y-intercept  =  1 The point of intersection is (1, 4).

Example 2 :

Solve the system of linear equations by substitution. Check your answer by graphing.

x + y  =  8

2x + y  =  11

Solution :

Step 1 :

Solve an equation for one variable.

Select one of the equation, say x + y  =  8.

Solve for the variable y in terms of x.

Subtract x from both sides.

(x + y) - x  =  (8) - x

x + y - x  =  8 - x

Simplify.

y  =  8 - x

Step 2 :

Substitute the expression for y in the other equation and solve.

2x + y  =  11

2x + (8 - x)  =  11

Combine like terms.

x + 8  =  11

Subtract 8 from both sides.

x  =  3

Step 3 :

Substitute the value of x we got above (x = 3) into one of the equations and solve for the other variable, y.

x + y  =  8

3 + y  =  8

Subtract 3 from both sides.

y  =  5

Hence, the solution of the system is (3, 5).

Step 4 :

Check the solution by graphing.

To graph the equations, write them in slope-intercept form.

That is,

y  =  mx + b

x + y  =  8

y  =  - x + 8

Slope  =  - 1

y-intercept  =  8

2x + y  =  11

y  =  -2x + 11

Slope  =  -2

Y-intercept  =  11 The point of intersection is (1, 4).

Example 3 :

A park charges \$10 for adults and \$5 for kids. How many many adults tickets and kids tickets were sold, if a total of 548 tickets were sold for a total of \$3750 ?

Solution :

Step 1 :

Let "x" be the number of adults tickets and "y" be the number of kids tickets.

No. of adults tickets + No. of kids tickets  =  Total

x + y  =  548 -----(2)

Step 2 :

Write an equation which represents the total cost.

Cost of "x" no. adults tickets  =  10x

Cost of "y" no. of kids tickets  =  5y

Total cost  =  \$3750

Then, we have

10x + 5y  =  3750

Divide both sides by 5.

2x + y  =  750 -----(2)

Step 3 :

Solve an equation for one variable.

Select one of the equation, say x + y  =  548.

Solve for the variable y in terms of x.

Subtract x from both sides.

(x + y) - x  =  (548) - x

y  =  548 - x

Step 4 :

Substitute the expression for y in the other equation and solve.

2x + y  =  750

2x + (548 - x)  =  750

Combine like terms.

x + 548  =  750

Subtract 548 from both sides.

x  =  202

Step 5 :

Substitute the value of x we got above (x = 202) into one of the equations and solve for the other variable, y.

x + y  =  548

202 + y  =  548

Subtract 202 from both sides.

y  =  346

So, the solution of the system is (202, 346).

Step 6 :

Interpret the solution in the original context.

So, the number of adults tickets sold is 202 and the number of kids tickets sold is 346. Apart from the stuff given above, if you need any other stuff in math, please use our google custom search here.

If you have any feedback about our math content, please mail us :

v4formath@gmail.com

You can also visit the following web pages on different stuff in math.

WORD PROBLEMS

Word problems on simple equations

Word problems on linear equations

Algebra word problems

Word problems on trains

Area and perimeter word problems

Word problems on direct variation and inverse variation

Word problems on unit price

Word problems on unit rate

Word problems on comparing rates

Converting customary units word problems

Converting metric units word problems

Word problems on simple interest

Word problems on compound interest

Word problems on types of angles

Complementary and supplementary angles word problems

Double facts word problems

Trigonometry word problems

Percentage word problems

Profit and loss word problems

Markup and markdown word problems

Decimal word problems

Word problems on fractions

Word problems on mixed fractrions

One step equation word problems

Linear inequalities word problems

Ratio and proportion word problems

Time and work word problems

Word problems on sets and venn diagrams

Word problems on ages

Pythagorean theorem word problems

Percent of a number word problems

Word problems on constant speed

Word problems on average speed

Word problems on sum of the angles of a triangle is 180 degree

OTHER TOPICS

Profit and loss shortcuts

Percentage shortcuts

Times table shortcuts

Time, speed and distance shortcuts

Ratio and proportion shortcuts

Domain and range of rational functions

Domain and range of rational functions with holes

Graphing rational functions

Graphing rational functions with holes

Converting repeating decimals in to fractions

Decimal representation of rational numbers

Finding square root using long division

L.C.M method to solve time and work problems

Translating the word problems in to algebraic expressions

Remainder when 2 power 256 is divided by 17

Remainder when 17 power 23 is divided by 16

Sum of all three digit numbers divisible by 6

Sum of all three digit numbers divisible by 7

Sum of all three digit numbers divisible by 8

Sum of all three digit numbers formed using 1, 3, 4

Sum of all three four digit numbers formed with non zero digits

Sum of all three four digit numbers formed using 0, 1, 2, 3

Sum of all three four digit numbers formed using 1, 2, 5, 6

1. Click on the HTML link code below.

Featured Categories

Math Word Problems

SAT Math Worksheet

P-SAT Preparation

Math Calculators

Quantitative Aptitude

Transformations

Algebraic Identities

Trig. Identities

SOHCAHTOA

Multiplication Tricks

PEMDAS Rule

Types of Angles

Aptitude Test
{"ft_lang_label":"__label__en","ft_lang_prob":0.80196416,"math_prob":0.99857527,"size":9098,"snap":"2021-43-2021-49","text_gpt3_token_len":2359,"char_repetition_ratio":0.15713657,"word_repetition_ratio":0.23658536,"special_character_ratio":0.24752693,"punctuation_ratio":0.07382134,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9999629,"pos_list":[0,1,2,3,4,5,6,7,8],"im_url_duplicate_count":[null,4,null,4,null,null,null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-12-04T23:30:08Z\",\"WARC-Record-ID\":\"<urn:uuid:ddabf7a6-7d07-462b-a31e-7bfbdabed35a>\",\"Content-Length\":\"102393\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:7eacd34d-1380-4678-974e-bc065a4706ed>\",\"WARC-Concurrent-To\":\"<urn:uuid:dd0ad1f3-c4f0-4e02-8678-0186427573e3>\",\"WARC-IP-Address\":\"173.247.218.242\",\"WARC-Target-URI\":\"https://www.onlinemath4all.com/solving-systems-by-substitution.html\",\"WARC-Payload-Digest\":\"sha1:HU462UWYSSRP7BZ4VJMICB4GO7APMZVS\",\"WARC-Block-Digest\":\"sha1:GLKXCSU6KQWGCMMEKPP3PYID7ARKRKTS\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-49/CC-MAIN-2021-49_segments_1637964363125.46_warc_CC-MAIN-20211204215252-20211205005252-00525.warc.gz\"}"}