2.3-Reducing Equations to Simpler Form

2.3-Reducing Equations to Simpler Form Important Formulae

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Grade 8 → Math → Linear Equations in One Variable → 2.3-Reducing Equations to Simpler Form

2.3-Reducing Equations to Simpler Form
  • Reducing equations involves simplifying both sides of the equation.
  • Combine like terms on both sides to make the equation simpler.
  • Example: $2x + 3x = 5x$ (Combine $2x$ and $3x$).
  • If there are constants on both sides, move them to one side using addition or subtraction.
  • Example: $2x + 5 = 3x + 7 \Rightarrow 2x - 3x = 7 - 5$.
  • Use distributive property if necessary: $a(b + c) = ab + ac$.
  • Always check if further simplification is possible after each step.

2.3-Reducing Equations to Simpler Form

In this section, we focus on simplifying linear equations in one variable. The goal is to reduce complex equations into simpler forms by applying basic algebraic rules and operations. This process makes solving linear equations easier and more efficient. Below are the key steps and techniques used to simplify equations:

1. Combine Like Terms

Like terms are terms that have the same variable raised to the same power. To simplify an equation, we first combine the like terms on both sides. For example:

3x + 5x = 8x

Similarly, constant terms (numbers without variables) can be combined:

7 + 3 = 10
2. Use the Distributive Property

If the equation has parentheses, apply the distributive property to remove them. The distributive property states that:

a(b + c) = ab + ac

For example:

2(3x + 4) = 6x + 8

Distribute the multiplication across the terms inside the parentheses to simplify the expression.

3. Move All Variables to One Side

To simplify an equation further, we aim to get all terms containing the variable on one side of the equation and constants on the other side. This is done by adding or subtracting terms from both sides. For example, if we have:

3x + 5 = 2x + 7

We can subtract $2x$ from both sides to get:

3x - 2x + 5 = 7

Which simplifies to:

x + 5 = 7
4. Simplify Coefficients

In some equations, terms may have coefficients. To simplify, divide or multiply both sides by the same number to reduce the coefficient of the variable. For example:

6x = 18

By dividing both sides by 6, we get:

x = 3
5. Eliminate Fractions

If an equation contains fractions, it is easier to simplify by eliminating them. Multiply both sides of the equation by the least common denominator (LCD) of all fractions. For example:

$\frac{1}{2}x + 3 = 7$

The least common denominator of 2 is 2. Multiply the entire equation by 2:

2 \times \left( \frac{1}{2}x + 3 \right) = 2 \times 7

This simplifies to:

x + 6 = 14
6. Isolate the Variable

Once the equation is simplified, the next step is to isolate the variable (usually represented as $x$) on one side of the equation. To do this, use addition, subtraction, multiplication, or division. For example, in the equation:

x + 6 = 14

Subtract 6 from both sides:

x = 14 - 6

Which simplifies to:

x = 8
7. Check the Simplified Equation

After simplifying the equation and solving for $x$, always check your solution by substituting the value of $x$ back into the original equation to verify that both sides are equal.

By applying these techniques, complex linear equations can be reduced to simpler forms, making it easier to solve for the variable.

Solve the following linear equations.

1. $\dfrac{x}{2} - \dfrac{1}{5} = \dfrac{x}{3} + \dfrac{1}{4}$

2. $\dfrac{n}{2} - \dfrac{3n}{4} + \dfrac{5n}{6} = 21$

3. $x + 7 - \dfrac{8x}{3} = \dfrac{17}{6} - \dfrac{5x}{2}$

4. $\dfrac{x - 5}{3} = \dfrac{x - 3}{5}$

5. $\dfrac{3t - 2}{4} - \dfrac{2t + 3}{3} = \dfrac{2}{3} - t $

6. $m - \dfrac{m - 1}{2} = 1 - \dfrac{m - 2}{3}$

Solution:

1. $\dfrac{x}{2} - \dfrac{1}{5} = \dfrac{x}{3} + \dfrac{1}{4}$
Step 1: Eliminate the fractions by multiplying the entire equation by 60 (LCM of 2, 5, 3, and 4).
$60 \times \left(\dfrac{x}{2} - \dfrac{1}{5} = \dfrac{x}{3} + \dfrac{1}{4}\right)$
$30x - 12 = 20x + 15$
Step 2: Simplify the equation.
$30x - 20x = 15 + 12$
$10x = 27$
Step 3: Solve for $x$.
$x = \dfrac{27}{10}$

2. $\dfrac{n}{2} - \dfrac{3n}{4} + \dfrac{5n}{6} = 21$
Step 1: Eliminate the fractions by multiplying the entire equation by 12 (LCM of 2, 4, and 6).
$12 \times \left(\dfrac{n}{2} - \dfrac{3n}{4} + \dfrac{5n}{6} = 21\right)$
$6n - 9n + 10n = 252$
Step 2: Simplify the equation.
$6n + 10n - 9n = 252$
$7n = 252$
Step 3: Solve for $n$.
$n = \dfrac{252}{7}$
$n = 36$

3. $x + 7 - \dfrac{8x}{3} = \dfrac{17}{6} - \dfrac{5x}{2}$
Step 1: Eliminate the fractions by multiplying the entire equation by 6 (LCM of 3, 6, and 2).
$6 \times \left(x + 7 - \dfrac{8x}{3} = \dfrac{17}{6} - \dfrac{5x}{2}\right)$
$6x + 42 - 16x = 17 - 15x$
Step 2: Simplify the equation.
$6x - 16x + 15x = 17 - 42$
$5x = -25$
Step 3: Solve for $x$.
$x = \dfrac{-25}{5}$
$x = -5$

4. $\dfrac{x - 5}{3} = \dfrac{x - 3}{5}$
Step 1: Eliminate the fractions by multiplying the entire equation by 15 (LCM of 3 and 5).
$15 \times \left(\dfrac{x - 5}{3} = \dfrac{x - 3}{5}\right)$
$5(x - 5) = 3(x - 3)$
Step 2: Simplify the equation.
$5x - 25 = 3x - 9$
Step 3: Move the terms involving $x$ to one side.
$5x - 3x = 25 - 9$
$2x = 16$
Step 4: Solve for $x$.
$x = \dfrac{16}{2}$
$x = 8$

5. $\dfrac{3t - 2}{4} - \dfrac{2t + 3}{3} = \dfrac{2}{3} - t $
Step 1: Eliminate the fractions by multiplying the entire equation by 12 (LCM of 4, 3).
$12 \times \left(\dfrac{3t - 2}{4} - \dfrac{2t + 3}{3} = \dfrac{2}{3} - t\right)$
$3(3t - 2) - 4(2t + 3) = 8 - 12t$
Step 2: Simplify the equation.
$9t - 6 - 8t - 12 = 8 - 12t$
Step 3: Combine like terms.
$t - 18 = 8 - 12t$
Step 4: Move the terms involving $t$ to one side.
$t + 12t = 8 + 18$
$13t = 26$
Step 5: Solve for $t$.
$t = \dfrac{26}{13}$
$t = 2$

6. $m - \dfrac{m - 1}{2} = 1 - \dfrac{m - 2}{3}$
Step 1: Eliminate the fractions by multiplying the entire equation by 6 (LCM of 2 and 3).
$6 \times \left(m - \dfrac{m - 1}{2} = 1 - \dfrac{m - 2}{3}\right)$
$6m - 3(m - 1) = 6 - 2(m - 2)$
Step 2: Simplify the equation.
$6m - 3m + 3 = 6 - 2m + 4$
Step 3: Combine like terms.
$3m + 3 = 10 - 2m$
Step 4: Move the terms involving $m$ to one side.
$3m + 2m = 10 - 3$
$5m = 7$
Step 5: Solve for $m$.
$m = \dfrac{7}{5}$
$m = 1.4$

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Simplify and solve the following linear equations.

7. 3(t–3)=5(2t+1)

8. 15(y–4)–2(y–9)+5(y+6)=0

9. 3(5z–7)–2(9z–11)=4(8z–13)–17

10. 0.25(4f – 3) = 0.05(10f – 9)

Solution:

7. Solve 3(t–3) = 5(2t+1)

$3(t-3) = 5(2t+1)$
$3t - 9 = 10t + 5$
$3t - 10t = 5 + 9$
$-7t = 14$
$t = \frac{14}{-7}$
$t = -2$

8. Solve 15(y–4)–2(y–9)+5(y+6) = 0

$15(y-4) - 2(y-9) + 5(y+6) = 0$
$15y - 60 - 2y + 18 + 5y + 30 = 0$
$15y - 2y + 5y = 60 - 18 - 30$
$18y = 12$
$y = \frac{12}{18}$
$y = \frac{2}{3}$

9. Solve 3(5z–7)–2(9z–11) = 4(8z–13)–17

$3(5z-7) - 2(9z-11) = 4(8z-13) - 17$
$15z - 21 - 18z + 22 = 32z - 52 - 17$
$15z - 18z = 32z - 52 - 17 + 21 - 22$
$-3z = 32z - 69$
$-3z - 32z = -69$
$-35z = -69$
$z = \frac{-69}{-35}$
$z = \frac{69}{35}$

10. Solve 0.25(4f – 3) = 0.05(10f – 9)

$0.25(4f - 3) = 0.05(10f - 9)$
$1f - 0.75 = 0.5f - 0.45$
$f - 0.5f = -0.45 + 0.75$
$0.5f = 0.3$
$f = \frac{0.3}{0.5}$
$f = 0.6$

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