Linear Equations — Algebra I Übungsblatt

Grades 8–9 Leichter Schwierigkeitsgrad 10 Aufgaben

A linear equation in one variable has the form $ax + b = c$ , where the variable appears only to the first power. Solving means finding the value of the variable that makes the equation true.

The golden rule of equation solving is balance: whatever you do to one side, do to the other. Work in reverse order of operations — undo addition/subtraction first, then multiplication/division.

When variables appear on both sides, collect them on one side and constants on the other. Always check your answer by substituting into the original equation.

Geübte Fähigkeiten

One-step and two-step equations
Equations with variables on both sides
Distributive property in equations
Checking solutions

Name: Datum: Stunde:

Übungsblatt: Linear Equations

Anleitung: Löse jede Aufgabe sorgfältig. Zeige deine Arbeit klar. Schreibe deine endgültige Antwort in das vorgesehene Feld oder wie angewiesen auf ein separates Blatt.

1.Solve for $x$ : $3x + 7 = 22$ .

Linear Equations — Übungsblatt (Fortsetzung)

2.Solve for $y$ : $5y - 9 = 2y + 6$ .
3.Solve for $a$ : $4(a - 3) = 2a + 8$ .

Linear Equations — Übungsblatt (Fortsetzung)

4.Solve for $b$ : $\dfrac{b}{5} + 2 = 7$ .
5.Solve for $c$ : $6c + 4 = 2c - 12$ .

Linear Equations — Übungsblatt (Fortsetzung)

6.A number increased by 9 is equal to 24. Find the number.
7.The sum of twice a number and 5 is 19. What is the number?

Linear Equations — Übungsblatt (Fortsetzung)

8.Three times a number, minus 4, is equal to 11. Find the number.
9.A rectangle has a length that is 3 inches more than its width. The perimeter is 30 inches. Find the width and length.

Linear Equations — Übungsblatt (Fortsetzung)

10.Tickets to a play cost $8 for adults and $5 for children. A group of 12 people paid $78 total. How many adults and how many children were in the group?

Lösungsschlüssel

1.

Subtract 7: $3x = 15$ ; Divide by 3: $x = 5$

Endgültige Antwort: $x = 5$
2.

Subtract $2y$ : $3y - 9 = 6$ ; Add 9: $3y = 15$ ; Divide by 3: $y = 5$

Endgültige Antwort: $y = 5$
3.

Distribute: $4a - 12 = 2a + 8$ ; Subtract $2a$ : $2a - 12 = 8$ ; Add 12: $2a = 20$ ; Divide by 2: $a = 10$

Endgültige Antwort: $a = 10$
4.

Subtract 2: $\dfrac{b}{5} = 5$ ; Multiply by 5: $b = 25$

Endgültige Antwort: $b = 25$
5.

Subtract $2c$ : $4c + 4 = -12$ ; Subtract 4: $4c = -16$ ; Divide by 4: $c = -4$

Endgültige Antwort: $c = -4$
6.

Let $n$ be the number. $n + 9 = 24$ ; Subtract 9: $n = 15$

Endgültige Antwort: The number is 15.

Lösungsschlüssel (Fortsetzung)

7.

Let $n$ be the number. $2n + 5 = 19$ ; Subtract 5: $2n = 14$ ; Divide by 2: $n = 7$

Endgültige Antwort: The number is 7.
8.

Let $n$ be the number. $3n - 4 = 11$ ; Add 4: $3n = 15$ ; Divide by 3: $n = 5$

Endgültige Antwort: The number is 5.
9.

Let $w$ be the width. Then length $= w + 3$ . Perimeter: $2w + 2(w + 3) = 30$ ; Simplify: $4w + 6 = 30$ ; Subtract 6: $4w = 24$ ; Divide by 4: $w = 6$ ; Length: $6 + 3 = 9$

Endgültige Antwort: Width = 6 inches, Length = 9 inches.
10.

Let $a$ be number of adults, $c$ be number of children. $a + c = 12$ ; $8a + 5c = 78$ ; From first: $c = 12 - a$ ; Substitute: $8a + 5(12 - a) = 78$ ; $3a + 60 = 78$ ; Subtract 60: $3a = 18$ ; Divide by 3: $a = 6$ ; Then $c = 12 - 6 = 6$

Endgültige Antwort: 6 adults and 6 children.

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Häufige Fehler, die es zu vermeiden gilt

Sign errors when moving terms across the equals sign
Dividing only one term instead of the entire side
Forgetting to distribute to every term inside parentheses
Accepting extraneous solutions without checking

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