Questions: Multi-Step Linear Equations (MA.8.AR.2.1) Solve each equation: a) 4.5 p-3=1.5 p+2(3 p+1) b) -2.3(n+1.5)+4.6 n=21.4 c) 0.35(m-2.4)+1.2=19.6

Multi-Step Linear Equations (MA.8.AR.2.1) Solve each equation: a) 4.5 p-3=1.5 p+2(3 p+1) b) -2.3(n+1.5)+4.6 n=21.4 c) 0.35(m-2.4)+1.2=19.6
Transcript text: B: Multi-Step Linear Equations (MA.8.AR.2.1) Solve each equation: a) $4.5 p-3=1.5 p+2(3 p+1)$ b) $-2.3(n+1.5)+4.6 n=21.4$ c) $0.35(m-2.4)+1.2=19.6$
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Solution

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Solution Steps

To solve each of these multi-step linear equations, we will first simplify both sides of the equation by distributing and combining like terms. Then, we will isolate the variable by moving all terms containing the variable to one side and constant terms to the other side. Finally, we will solve for the variable by dividing or multiplying as necessary.

Step 1: Solve Equation a

We start with the equation: \[ 4.5p - 3 = 1.5p + 2(3p + 1) \] First, we simplify the right side: \[ 2(3p + 1) = 6p + 2 \] Thus, the equation becomes: \[ 4.5p - 3 = 1.5p + 6p + 2 \] Combining like terms gives: \[ 4.5p - 3 = 7.5p + 2 \] Next, we isolate \(p\): \[ 4.5p - 7.5p = 2 + 3 \] This simplifies to: \[ -3p = 5 \implies p = -\frac{5}{3} \approx -1.6667 \]

Step 2: Solve Equation b

We start with the equation: \[ -2.3(n + 1.5) + 4.6n = 21.4 \] Distributing \(-2.3\) gives: \[ -2.3n - 3.45 + 4.6n = 21.4 \] Combining like terms results in: \[ (4.6n - 2.3n) - 3.45 = 21.4 \] This simplifies to: \[ 2.3n - 3.45 = 21.4 \] Isolating \(n\): \[ 2.3n = 21.4 + 3.45 \] Thus: \[ 2.3n = 24.85 \implies n = \frac{24.85}{2.3} \approx 10.8043 \]

Step 3: Solve Equation c

We start with the equation: \[ 0.35(m - 2.4) + 1.2 = 19.6 \] Distributing \(0.35\) gives: \[ 0.35m - 0.84 + 1.2 = 19.6 \] Combining like terms results in: \[ 0.35m + 0.36 = 19.6 \] Isolating \(m\): \[ 0.35m = 19.6 - 0.36 \] Thus: \[ 0.35m = 19.24 \implies m = \frac{19.24}{0.35} \approx 54.9714 \]

Final Answer

The solutions to the equations are:

  • For \(p\): \(\boxed{p = -\frac{5}{3}}\)
  • For \(n\): \(\boxed{n \approx 10.8043}\)
  • For \(m\): \(\boxed{m \approx 54.9714}\)
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