Level: Intermediate
- = ADP processing week number (Sunday – Thursday) If you make a schedule change, please check your Payroll Schedule to be sure you use the correct week number.
- 驚きの収納力と価格以上の機能性 シンプルで使いやすいバッグ トートバッグ メンズ Velba Success(ベルバサクセス) ATシリーズ トートバック 大きめ 9605。トートバッグ メンズ 布 ブランド 大きめ ビジネス Velba Success ベルバサクセス ATシリーズ ナイロン 2way A4 ノートPC対応 ショルダーバッグ.
- It’s as quick as Mac OSX’s native preview or PhotoMechanic, but it offers much more useful video controls. In fact, it’s the way that Fileloupe lets you view and sort through GoPro video footage that is the real killer feature for me. And finally, at $14.99, Fileloupe is priced much more attractively than some of the other apps.
Jeremiah 31:7-14. Eph 1:3-6, 15-19a. Matt 2:13-15, 19-23 or Luke 2:41-52 or Matt 2:1-12. January 12, 2020, Baptism of the Lord, 1st Sunday of Epiphany, 1st Sun of Ord.
- Torah is divided into 54 portions for weekly reading in synagogue
- There are also special readings for holidays and other days
![Kjv Kjv](https://havecamerawilltravel.com/photographer/files/2015/01/lighroom-alternatives.jpg)
Each week in synagogue, we read (or, more accurately, chant, because it is sung) a passage from the Torah. This passage is referred to as a parshah. The first parshah, for example, is Parshat Bereishit, which covers from the beginning of Genesis to the story of Noah. There are 54 parshahs, one for each week of a leap year, so that in the course of a year, we read the entire Torah (Genesis to Deuteronomy) in our services. During non-leap years, there are 50 weeks, so some of the shorter portions are doubled up. We read the last portion of the Torah right before a holiday called Simchat Torah (Rejoicing in the Law), which occurs in October, a few weeks after Rosh Hashanah (Jewish New Year). On Simchat Torah, we read the last portion of the Torah, and proceed immediately to the first paragraph of Genesis, showing that the Torah is a circle, and never ends.
In the synagogue service, the weekly parshah is followed by a passage from the prophets, which is referred to as a haftarah. Contrary to common misconception, 'haftarah' does not mean 'half-Torah.' The word comes from the Hebrew root Fei-Teit-Reish and means 'Concluding Portion'. Usually, haftarah portion is no longer than one chapter, and has some relation to the Torah portion of the week.
The Torah and haftarah readings are performed with great ceremony: the Torah is paraded around the room before it is brought to rest on the bimah (podium). The reading is divided up into portions, and various members of the congregation have the honor of reciting a blessing over a portion of the reading. This honor is referred to as an aliyah (literally, ascension).
The first aliyah of any day's reading is reserved for a kohein, the second for a Levite, and priority for subsequent aliyot are given to people celebrating major life events, such as marriage or the birth of a child. In fact, a Bar Mitzvah was originally nothing more than the first aliyah of a boy who had reached the age to be permitted such an honor. Celebrants of life events are ordinarily given the last aliyah, which includes blessing the last part of the Torah reading as well as blessing the haftarah reading. The person given this honor is referred to as the maftir, from the same root as haftarah, meaning 'the one who concludes.'
For more information about services, see Jewish Liturgy.
Jewish scriptures are sometimes bound in a form that corresponds to this division into weekly readings. Scriptures bound in this way are generally referred to as a chumash. The word 'chumash' comes from the Hebrew word meaning five, and refers to the five books of the Torah. Sometimes, a chumash is simply refers to a collection of the five books of the Torah. But often, a chumash contains the entire first five books, divided up by the weekly parshiyot, with the haftarah portion for each week inserted immediately after the week's parshah.
Below is a table of the regular weekly scriptural readings. Haftarot in parentheses indicate Sephardic ritual where it differs from Ashkenazic. There are other variations on the readings, but these are the most commonly used ones. If you want to know the reading for this week, check the Current Calendar.
There are additional special readings for certain holidays and other special days, listed in a separate table below.
- Parshah
- Torah
- Haftarah
- Bereishit
- Genesis 1:1-6:8
- Isaiah 42:5-43:11
(Isaiah 42:5-42:21) - Noach
- Genesis 6:9-11:32
- Isaiah 54:1-55:5
(Isaiah 54:1-10) - Lekh Lekha
- Genesis 12:1-17:27
- Isaiah 40:27-41:16
- Vayeira
- Genesis 18:1-22:24
- II Kings 4:1-4:37
(II Kings 4:1-4:23) - Chayei Sarah
- Genesis 23:1-25:18
- I Kings1:1-1:31
- Toldot
- Genesis 25:19-28:9
- Malachi 1:1-2:7
- Vayeitzei
- Genesis 28:10-32:3
- Hosea 12:13-14:10
(Hosea 11:7-12:12) - Vayishlach
- Genesis 32:4-36:43
- Hosea 11:7-12:12
(Obadiah1:1-1:21) - Vayyeshev
- Genesis 37:1-40:23
- Amos 2:6-3:8
- Miqeitz
- Genesis 41:1-44:17
- I Kings 3:15-4:1
- Vayigash
- Genesis 44:18-47:27
- Ezekiel 37:15-37:28
- Vayechi
- Genesis 47:28-50:26
- I Kings 2:1-12
- Shemot
- Exodus 1:1-6:1
- Isaiah 27:6-28:13; 29:22-29:23
(Jeremiah 1:1-2:3) - Va'eira
- Exodus 6:2-9:35
- Ezekiel 28:25-29:21
- Bo
- Exodus 10:1-13:16
- Jeremiah 46:13-46:28
- Beshalach (Shabbat Shirah)
- Exodus 13:17-17:16
- Judges 4:4-5:31
(Judges 5:1-5:31) - Yitro
- Exodus 18:1-20:23
- Isaiah 6:1-7:6; 9:5-9:6
(Isaiah 6:1-6:13) - Mishpatim
- Exodus 21:1-24:18
- Jeremiah 34:8-34:22; 33:25-33:26
- Terumah
- Exodus 25:1-27:19
- I Kings 5:26-6:13
- Tetzaveh
- Exodus 27:20-30:10
- Ezekiel 43:10-43:27
- Ki Tisa
- Exodus 30:11-34:35
- I Kings 18:1-18:39
(I Kings 18:20-18:39) - Vayaqhel
- Exodus 35:1-38:20
- I Kings 7:40-7:50
(I Kings 7:13-7:26) - Pequdei
- Exodus 38:21-40:38
- I Kings 7:51-8:21
(I Kings 7:40-7:50) - Vayiqra
- Leviticus 1:1-5:26
- Isaiah 43:21-44:23
- Tzav
- Leviticus 6:1-8:36
- Jeremiah 7:21-8:3; 9:22-9:23
- Shemini
- Leviticus 9:1-11:47
- II Samuel 6:1-7:17
(II Samuel 6:1-6:19) - Tazria
- Leviticus 12:1-13:59
- II Kings 4:42-5:19
- Metzora
- Leviticus 14:1-15:33
- II Kings 7:3-7:20
- Acharei Mot
- Leviticus 16:1-18:30
- Ezekiel 22:1-22:19
(Ezekiel 22:1-22:16) - Qedoshim
- Leviticus 19:1-20:27
- Amos 9:7-9:15
(Ezekiel 20:2-20:20) - Emor
- Leviticus 21:1-24:23
- Ezekiel 44:15-44:31
- Behar
- Leviticus 25:1-26:2
- Jeremiah 32:6-32:27
- Bechuqotai
- Leviticus 26:3-27:34
- Jeremiah 16:19-17:14
- Bamidbar
- Numbers 1:1-4:20
- Hosea 2:1-2:22
- Nasso
- Numbers 4:21-7:89
- Judges 13:2-13:25
- Beha'alotkha
- Numbers 8:1-12:16
- Zechariah 2:14-4:7
- Shelach
- Numbers 13:1-15:41
- Joshua 2:1-2:24
- Qorach
- Numbers 16:1-18:32
- I Samuel 11:14-12:22
- Chuqat
- Numbers 19:1-22:1
- Judges 11:1-11:33
- Balaq
- Numbers 22:2-25:9
- Micah 5:6-6:8
- Pinchas
- Numbers 25:10-30:1
- I Kings 18:46-19:21
- Mattot
- Numbers 30:2-32:42
- Jeremiah 1:1-2:3
- Masei
- Numbers 33:1-36:13
- Jeremiah 2:4-28; 3:4
(Jeremiah 2:4-28; 4:1-4:2) - Devarim
- Deuteronomy 1:1-3:22
- Isaiah 1:1-1:27
- Va'etchanan
- Deuteronomy 3:23-7:11
- Isaiah 40:1-40:26
- Eiqev
- Deuteronomy 7:12-11:25
- Isaiah 49:14-51:3
- Re'eh
- Deuteronomy 11:26-16:17
- Isaiah 54:11-55:5
- Shoftim
- Deuteronomy 16:18-21:9
- Isaiah 51:12-52:12
- Ki Teitzei
- Deuteronomy 21:10-25:19
- Isaiah 54:1-54:10
- Ki Tavo
- Deuteronomy 26:1-29:8
- Isaiah 60:1-60:22
- Nitzavim
- Deuteronomy 29:9-30:20
- Isaiah 61:10-63:9
- Vayeilekh
- Deuteronomy 31:1-31:30
- Isaiah 55:6-56:8
- Ha'azinu
- Deuteronomy 32:1-32:52
- II Samuel 22:1-22:51
- Vezot Haberakhah
- Deuteronomy 33:1-34:12
- Joshua 1:1-1:18
(Joshua 1:1-1:9)
Below are additional readings for holidays and special Shabbats. Haftarot in parentheses indicate Sephardic ritual where it differs from Ashkenazic. Note that on holidays, the Maftir portion ordinarily comes from a different Torah scroll. The Maftir portion is usually the Torah portion that institutes the holiday or specifies the holiday's offerings.
- Parshah
- Torah
- Maftir
- Haftarah
- Rosh Hashanah, Day 1
- Gen 21:1-34
- Num 29:1-6
- I Sam 1:1-2:10
- Rosh Hashanah, Day 2
- Gen 22:1-24
- Num 29:1-6
- Jer 31:1-19
- Shabbat Shuvah
- Hosea 14,2-10; Joel 2,15-27
(Hosea 14,2-10; Micah 7,18-20) - Yom Kippur, Morning
- Lev 16:1-34
- Num 29:7-11
- Is 57:14-58:14
- Yom Kippur, Afternoon
- Lev 18:1-30
- Jonah 1:1-4:11
Micah 7:18-20 - Sukkot, Day 1
- Lev 22:26-23:44
- Num 29:12-16
- Zech 14:1-21
- Sukkot, Day 2
- Lev 22:26-23:44
- Num 29:12-16
- I Kings 8:2-21
- Sukkot, Intermediate Shabbat
- Ex 33:12-34:26
- Ezek 38:18-39:16
- Sukkot, Chol Ha-mo'ed Day 1
- Num 29:17-25
- Sukkot, Chol Ha-mo'ed Day 2
- Num 29:20-28
- Sukkot, Chol Ha-mo'ed Day 3
- Num 29:23-31
- SukkotChol Ha-mo'ed Day 4
- Num 29:26-34
- Hoshanah Rabbah(Sukkot, Day 7)
- Num 29:26-34
- Shemini Atzeret
- Deut 14:22-16:17
- Num 29:35-30:1
- I Ki 8:54-9:1
- Simchat Torah
- Deut 33:1-34:12
Gen 1:1-2:3 - Num 29:35-30:1
- Josh 1:1-18
(Josh 1:1-9) - Chanukkah, Day 1
- Num 7:1-17
- Chanukkah, Day 2
- Num 7:18-29
- Chanukkah, Day 3
- Num 7:24-35
- Chanukkah, Day 4
- Num 7:30-41
- Chanukkah, Day 5
- Num 7:36-47
- Chanukkah, Day 6 (if Rosh Chodesh)
- Num 28:1-15
- Num 7:42-47
- Chanukkah, Day 7 (if Rosh Chodesh)
- Num 28:1-15
- Num 7:48-59
- Chanukkah, Day 7 (if not Rosh Chodesh)
- Num 7:48-59
- Chanukkah, Day 8
- Num 7:54-8:4
- Chanukkah, First Intermediate Shabbat
- Zechariah 2:14-4:7
- Chanukkah, Second Intermediate Shabbat
- 1 Kings 7:40-50
- Sheqalim
- Ex 30:11-16
- II Ki 12:1-17
(II Ki 11:17-12:17) - Zakhor
- Deut 25:17-19
- I Sam 15:2-34
(I Sam 15:1-34) - Purim
- Ex 17:8-16
- Parah
- Num 19:1-22
- Ezek 36:16-38
(Ezek 36:16-36) - Ha-Chodesh
- Ex 12:1-20
- Ezek 45:16-46:18
(Ezek 45:18-46:18) - Shabbat Ha-Gadol
- Mal 3:4-24
- Pesach (Passover), Day 1
- Ex12:21-51
- Num 28:16-25
- Josh3:5-7; 5:2-6:1; 6:27
(Josh 5:2-6:1) - Pesach (Passover), Day 2
- Lev 22:26-23:44
- Num 28:16-25
- II Ki 23:1-9; 21-25
- Pesach (Passover)
Intermediate Shabbat - Ex 33:12-34:26
- Num 28:19-25
- Ezek 37:1-37:14
(Ezek 36:37-37:14) - Pesach (Passover), Chol Ha-mo'ed Day 1
- Ex 13:1-16;
- Num 28:19-25
- Pesach (Passover), Chol Ha-mo'ed Day 2
- Ex 22:24-23:19;
- Num 28:19-25
- Pesach (Passover), Chol Ha-mo'ed Day 3
- Ex 34:1-26;
- Num 28:19-25
- Pesach (Passover), Chol Ha-mo'ed Day 4
- Num 9:1-14;
- Num 28:19-25
- Pesach (Passover), Day 7
- Ex 13:17-15:26
- Num 28:19-25
- II Sam 22:1-51
- Pesach (Passover), Day 8 (if weekday)
- Deut 15:19-16:17
- Num 28:19-25
- Is 10:32-12:6
- Pesach (Passover), Day 8 (if Shabbat)
- Deut 14:22-16:17
- Num 28:19-25
- Is 10:32-12:6
- Shavu'ot, Day 1
- Ex 19:1-20:23
- Num 28:26-31
- Ezek 1:1-28; 3:12
- Shavu'ot, Day 2 (if weekday)
- Deut 15:19-16:17
- Num 28:26-31
- Hab 2:20-3:19
- Shavu'ot, Day 2 (if Shabbat)
- Deut 14:22-16:17
- Num 28:26-31
- Hab 2:20-3:19
- Tisha B'Av, Morning
- Deut 4:25-40
- Jer 8:13-9:23
- Tisha B'Av, Afternoon
- Ex 32:11-14, 34:1-10
- Isaiah 55:6-56:8
(Hosea 14:2-10; Micah 7:18-20) - Minor Fasts, Morning
- Ex 32:11-14; 34:1-10
- Minor Fasts, Afternoon
- Ex 32:11-14; 34:1-10
- Is 55:6-56:8
(none) - Shabbat the day before Rosh Chodesh)
- I Sam 20:18-42
- Rosh Chodesh (weekday)
- Num 28:1-15
- Rosh Chodesh (Shabbat)
- Num 28:9-15
- Is 66:1-24
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What's Nu? | Current Calendar | About
Enter an equation along with the variable you wish to solve it for and click the Solve button.
In this chapter, we will develop certain techniques that help solve problems stated in words. These techniques involve rewriting problems in the form of symbols. For example, the stated problem
'Find a number which, when added to 3, yields 7'
may be written as:
3 + ? = 7, 3 + n = 7, 3 + x = 1
and so on, where the symbols ?, n, and x represent the number we want to find. We call such shorthand versions of stated problems equations, or symbolic sentences. Equations such as x + 3 = 7 are first-degree equations, since the variable has an exponent of 1. The terms to the left of an equals sign make up the left-hand member of the equation; those to the right make up the right-hand member. Thus, in the equation x + 3 = 7, the left-hand member is x + 3 and the right-hand member is 7.
SOLVING EQUATIONS
Equations may be true or false, just as word sentences may be true or false. The equation:
3 + x = 7
will be false if any number except 4 is substituted for the variable. The value of the variable for which the equation is true (4 in this example) is called the solution of the equation. We can determine whether or not a given number is a solution of a given equation by substituting the number in place of the variable and determining the truth or falsity of the result.
Example 1 Determine if the value 3 is a solution of the equation
4x - 2 = 3x + 1
Solution We substitute the value 3 for x in the equation and see if the left-hand member equals the right-hand member.
4(3) - 2 = 3(3) + 1
12 - 2 = 9 + 1
10 = 10
Ans. 3 is a solution.
The first-degree equations that we consider in this chapter have at most one solution. The solutions to many such equations can be determined by inspection.
Example 2 Find the solution of each equation by inspection.
a. x + 5 = 12
b. 4 · x = -20
b. 4 · x = -20
Solutions a. 7 is the solution since 7 + 5 = 12.
b. -5 is the solution since 4(-5) = -20.
b. -5 is the solution since 4(-5) = -20.
SOLVING EQUATIONS USING ADDITION AND SUBTRACTION PROPERTIES
In Section 3.1 we solved some simple first-degree equations by inspection. However, the solutions of most equations are not immediately evident by inspection. Hence, we need some mathematical 'tools' for solving equations.
EQUIVALENT EQUATIONS
Equivalent equations are equations that have identical solutions. Thus,
3x + 3 = x + 13, 3x = x + 10, 2x = 10, and x = 5
are equivalent equations, because 5 is the only solution of each of them. Notice in the equation 3x + 3 = x + 13, the solution 5 is not evident by inspection but in the equation x = 5, the solution 5 is evident by inspection. In solving any equation, we transform a given equation whose solution may not be obvious to an equivalent equation whose solution is easily noted.
The following property, sometimes called the addition-subtraction property, is one way that we can generate equivalent equations.
If the same quantity is added to or subtracted from both membersof an equation, the resulting equation is equivalent to the originalequation.
In symbols,
a - b, a + c = b + c, and a - c = b - c
are equivalent equations.
Example 1 Write an equation equivalent to
x + 3 = 7
by subtracting 3 from each member.
Solution Subtracting 3 from each member yields
x + 3 - 3 = 7 - 3
or
x = 4
Notice that x + 3 = 7 and x = 4 are equivalent equations since the solution is the same for both, namely 4. The next example shows how we can generate equivalent equations by first simplifying one or both members of an equation.
Example 2 Write an equation equivalent to
4x- 2-3x = 4 + 6
by combining like terms and then by adding 2 to each member.
Combining like terms yields
x - 2 = 10
Adding 2 to each member yields
Sketch 3 2 – vector drawing application. x-2+2 =10+2
x = 12
To solve an equation, we use the addition-subtraction property to transform a given equation to an equivalent equation of the form x = a, from which we can find the solution by inspection.
Example 3 Solve 2x + 1 = x - 2.
We want to obtain an equivalent equation in which all terms containing x are in one member and all terms not containing x are in the other. If we first add -1 to (or subtract 1 from) each member, we get
2x + 1- 1 = x - 2- 1
2x = x - 3
If we now add -x to (or subtract x from) each member, we get
2x-x = x - 3 - x
x = -3
where the solution -3 is obvious.
The solution of the original equation is the number -3; however, the answer is often displayed in the form of the equation x = -3.
Since each equation obtained in the process is equivalent to the original equation, -3 is also a solution of 2x + 1 = x - 2. In the above example, we can check the solution by substituting - 3 for x in the original equation
2(-3) + 1 = (-3) - 2
-5 = -5
The symmetric property of equality is also helpful in the solution of equations. This property states
If a = b then b = a
This enables us to interchange the members of an equation whenever we please without having to be concerned with any changes of sign. Thus,
If 4 = x + 2 then x + 2 = 4
If x + 3 = 2x - 5 then 2x - 5 = x + 3
If d = rt then rt = d
There may be several different ways to apply the addition property above. Sometimes one method is better than another, and in some cases, the symmetric property of equality is also helpful.
Example 4 Solve 2x = 3x - 9. (1)
Solution If we first add -3x to each member, we get
2x - 3x = 3x - 9 - 3x
-x = -9
where the variable has a negative coefficient. Although we can see by inspection that the solution is 9, because -(9) = -9, we can avoid the negative coefficient by adding -2x and +9 to each member of Equation (1). In this case, we get
2x-2x + 9 = 3x- 9-2x+ 9
9 = x
from which the solution 9 is obvious. If we wish, we can write the last equation as x = 9 by the symmetric property of equality.
SOLVING EQUATIONS USING THE DIVISION PROPERTY
Consider the equation
3x = 12
The solution to this equation is 4. Also, note that if we divide each member of the equation by 3, we obtain the equations
whose solution is also 4. In general, we have the following property, which is sometimes called the division property.
If both members of an equation are divided by the same (nonzero)quantity, the resulting equation is equivalent to the original equation.
In symbols,
are equivalent equations.
Example 1 Write an equation equivalent to
-4x = 12
by dividing each member by -4.
Solution Dividing both members by -4 yields
In solving equations, we use the above property to produce equivalent equations in which the variable has a coefficient of 1.
Example 2 Solve 3y + 2y = 20.
We first combine like terms to get
5y = 20
Then, dividing each member by 5, we obtain
In the next example, we use the addition-subtraction property and the division property to solve an equation.
Example 3 Solve 4x + 7 = x - 2.
Solution First, we add -x and -7 to each member to get
4x + 7 - x - 7 = x - 2 - x - 1
Next, combining like terms yields
3x = -9
Last, we divide each member by 3 to obtain
SOLVING EQUATIONS USING THE MULTIPLICATION PROPERTY
Consider the equation
The solution to this equation is 12. Also, note that if we multiply each member of the equation by 4, we obtain the equations
whose solution is also 12. In general, we have the following property, which is sometimes called the multiplication property.
If both members of an equation are multiplied by the same nonzero quantity, the resulting equation Is equivalent to the original equation.
In symbols,
a = b and a·c = b·c (c ≠ 0)
are equivalent equations.
Example 1 Write an equivalent equation to
by multiplying each member by 6.
Solution Multiplying each member by 6 yields
In solving equations, we use the above property to produce equivalent equations that are free of fractions.
Example 2 Solve
Solution First, multiply each member by 5 to get
Now, divide each member by 3,
Example 3 Solve .
Solution First, simplify above the fraction bar to get
Next, multiply each member by 3 to obtain
Last, dividing each member by 5 yields
FURTHER SOLUTIONS OF EQUATIONS
Now we know all the techniques needed to solve most first-degree equations. There is no specific order in which the properties should be applied. Any one or more of the following steps listed on page 102 may be appropriate.
Steps to solve first-degree equations:
- Combine like terms in each member of an equation.
- Using the addition or subtraction property, write the equation with all terms containing the unknown in one member and all terms not containing the unknown in the other.
- Combine like terms in each member.
- Use the multiplication property to remove fractions.
- Use the division property to obtain a coefficient of 1 for the variable.
Example 1 Solve 5x - 7 = 2x - 4x + 14.
![Fileloupe 1 7 14 Fileloupe 1 7 14](https://pbs.twimg.com/media/C7mA355VQAA7R42.png)
Solution First, we combine like terms, 2x - 4x, to yield
5x - 7 = -2x + 14
Next, we add +2x and +7 to each member and combine like terms to get
5x - 7 + 2x + 7 = -2x + 14 + 2x + 1
7x = 21
Finally, we divide each member by 7 to obtain
In the next example, we simplify above the fraction bar before applying the properties that we have been studying.
Example 2 Solve
Solution First, we combine like terms, 4x - 2x, to get
Then we add -3 to each member and simplify
Next, we multiply each member by 3 to obtain
Finally, we divide each member by 2 to get
SOLVING FORMULAS
Equations that involve variables for the measures of two or more physical quantities are called formulas. We can solve for any one of the variables in a formula if the values of the other variables are known. We substitute the known values in the formula and solve for the unknown variable by the methods we used in the preceding sections.
Example 1 In the formula d = rt, find t if d = 24 and r = 3.
Solution We can solve for t by substituting 24 for d and 3 for r. That is,
d = rt
(24) = (3)t
8 = t
It is often necessary to solve formulas or equations in which there is more than one variable for one of the variables in terms of the others. We use the same methods demonstrated in the preceding sections.
Example 2 In the formula d = rt, solve for t in terms of r and d.
Solution We may solve for t in terms of r and d by dividing both members by r to yield
from which, by the symmetric law,
In the above example, we solved for t by applying the division property to generate an equivalent equation. Sometimes, it is necessary to apply more than one such property.
Example 3 In the equation ax + b = c, solve for x in terms of a, b and c.
Fileloupe 1 7 14 Commentary
Solution We can solve for x by first adding -b to each member to get
Fileloupe 1 7 14 Mm
then dividing each member by a, we have