7th Grade Algebraic Expressions Worksheets - Math Monks - Free Printable

To solve the given problems, we will evaluate each algebraic expression step by step using the provided values for the variables. Let's go through each problem:

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Problem 1:

Evaluate \( \frac{3}{s - t} - 1 \) at \( s = 2 \), \( t = -6 \).

1. Substitute \( s = 2 \) and \( t = -6 \):
\[
\frac{3}{s - t} - 1 = \frac{3}{2 - (-6)} - 1
\]
2. Simplify the denominator:
\[
2 - (-6) = 2 + 6 = 8
\]
3. Substitute back:
\[
\frac{3}{8} - 1
\]
4. Convert 1 to a fraction with a denominator of 8:
\[
\frac{3}{8} - \frac{8}{8} = \frac{3 - 8}{8} = \frac{-5}{8}
\]

Answer:
\[
\boxed{-\frac{5}{8}}
\]

---

Problem 2:

Evaluate \( \frac{x + 1}{y^2} \) at \( x = 8 \), \( y = \frac{3}{5} \).

1. Substitute \( x = 8 \) and \( y = \frac{3}{5} \):
\[
\frac{x + 1}{y^2} = \frac{8 + 1}{\left(\frac{3}{5}\right)^2}
\]
2. Simplify the numerator:
\[
8 + 1 = 9
\]
3. Simplify the denominator:
\[
\left(\frac{3}{5}\right)^2 = \frac{3^2}{5^2} = \frac{9}{25}
\]
4. Substitute back:
\[
\frac{9}{\frac{9}{25}} = 9 \cdot \frac{25}{9} = 25
\]

Answer:
\[
\boxed{25}
\]

---

Problem 3:

Evaluate \( \frac{3}{u}(u^2 + 68) \) at \( u = -12 \).

1. Substitute \( u = -12 \):
\[
\frac{3}{u}(u^2 + 68) = \frac{3}{-12}((-12)^2 + 68)
\]
2. Simplify \( \frac{3}{-12} \):
\[
\frac{3}{-12} = -\frac{1}{4}
\]
3. Simplify \( (-12)^2 + 68 \):
\[
(-12)^2 = 144 \quad \text{and} \quad 144 + 68 = 212
\]
4. Substitute back:
\[
-\frac{1}{4} \cdot 212 = -\frac{212}{4} = -53
\]

Answer:
\[
\boxed{-53}
\]

---

Problem 4:

Evaluate \( 21r(63r + 3) \) at \( r = -\frac{3}{7} \).

1. Substitute \( r = -\frac{3}{7} \):
\[
21r(63r + 3) = 21\left(-\frac{3}{7}\right)\left(63\left(-\frac{3}{7}\right) + 3\right)
\]
2. Simplify \( 63r \):
\[
63r = 63 \cdot \left(-\frac{3}{7}\right) = -\frac{189}{7} = -27
\]
3. Simplify \( 63r + 3 \):
\[
63r + 3 = -27 + 3 = -24
\]
4. Substitute back:
\[
21\left(-\frac{3}{7}\right)(-24) = 21 \cdot \left(-\frac{3}{7}\right) \cdot (-24)
\]
5. Simplify step-by-step:
\[
21 \cdot \left(-\frac{3}{7}\right) = -\frac{63}{7} = -9
\]
\[
-9 \cdot (-24) = 216
\]

Answer:
\[
\boxed{216}
\]

---

Problem 5:

Evaluate \( 6n^2 + 4n^2 - 19 \) at \( n = -3 \).

1. Combine like terms:
\[
6n^2 + 4n^2 - 19 = 10n^2 - 19
\]
2. Substitute \( n = -3 \):
\[
10n^2 - 19 = 10(-3)^2 - 19
\]
3. Simplify \( (-3)^2 \):
\[
(-3)^2 = 9
\]
4. Substitute back:
\[
10 \cdot 9 - 19 = 90 - 19 = 71
\]

Answer:
\[
\boxed{71}
\]

---

Problem 6:

Evaluate \( \frac{12q - 34}{4} \) at \( q = -\frac{1}{3} \).

1. Substitute \( q = -\frac{1}{3} \):
\[
\frac{12q - 34}{4} = \frac{12\left(-\frac{1}{3}\right) - 34}{4}
\]
2. Simplify \( 12q \):
\[
12\left(-\frac{1}{3}\right) = -4
\]
3. Substitute back:
\[
\frac{-4 - 34}{4} = \frac{-38}{4} = -\frac{19}{2}
\]

Answer:
\[
\boxed{-\frac{19}{2}}
\]

---

Problem 7:

Evaluate \( \frac{5t}{3t - s} - 1 \) at \( s = 2 \), \( t = -6 \).

1. Substitute \( s = 2 \) and \( t = -6 \):
\[
\frac{5t}{3t - s} - 1 = \frac{5(-6)}{3(-6) - 2} - 1
\]
2. Simplify the numerator:
\[
5(-6) = -30
\]
3. Simplify the denominator:
\[
3(-6) - 2 = -18 - 2 = -20
\]
4. Substitute back:
\[
\frac{-30}{-20} - 1 = \frac{30}{20} - 1 = \frac{3}{2} - 1
\]
5. Convert 1 to a fraction with a denominator of 2:
\[
\frac{3}{2} - \frac{2}{2} = \frac{3 - 2}{2} = \frac{1}{2}
\]

Answer:
\[
\boxed{\frac{1}{2}}
\]

---

Problem 8:

Evaluate \( \frac{s - 4}{t^2} \) at \( s = 5 \), \( t = -5 \).

1. Substitute \( s = 5 \) and \( t = -5 \):
\[
\frac{s - 4}{t^2} = \frac{5 - 4}{(-5)^2}
\]
2. Simplify the numerator:
\[
5 - 4 = 1
\]
3. Simplify the denominator:
\[
(-5)^2 = 25
\]
4. Substitute back:
\[
\frac{1}{25}
\]

Answer:
\[
\boxed{\frac{1}{25}}
\]

---

Problem 9:

Evaluate \( \frac{6q - g}{4} \) at \( q = -4 \), \( g = 1 \).

1. Substitute \( q = -4 \) and \( g = 1 \):
\[
\frac{6q - g}{4} = \frac{6(-4) - 1}{4}
\]
2. Simplify \( 6q \):
\[
6(-4) = -24
\]
3. Substitute back:
\[
\frac{-24 - 1}{4} = \frac{-25}{4}
\]

Answer:
\[
\boxed{-\frac{25}{4}}
\]

---

Problem 10:

Evaluate \( (10p - 4)^2 \) at \( p = 2 \).

1. Substitute \( p = 2 \):
\[
(10p - 4)^2 = (10(2) - 4)^2
\]
2. Simplify \( 10p \):
\[
10(2) = 20
\]
3. Substitute back:
\[
(20 - 4)^2 = 16^2
\]
4. Simplify:
\[
16^2 = 256
\]

Answer:
\[
\boxed{256}
\]

---

Final Answers:

\[
\boxed{
\begin{aligned}
1. & \ -\frac{5}{8} \\
2. & \ 25 \\
3. & \ -53 \\
4. & \ 216 \\
5. & \ 71 \\
6. & \ -\frac{19}{2} \\
7. & \ \frac{1}{2} \\
8. & \ \frac{1}{25} \\
9. & \ -\frac{25}{4} \\
10. & \ 256
\end{aligned}
}
\]