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Chapter 3: Problem 32

Solve each equation. $$-16 t+3=67$$

Short Answer

Expert verified
The solution is \(t = -4\).

Step by step solution

01

Isolate the Variable Term

To solve for the variable \(t\), start by isolating the term containing \(t\) on one side of the equation. Subtract 3 from both sides to eliminate the constant on the left side of the equation: \[-16t + 3 - 3 = 67 - 3\] This simplifies to: \[-16t = 64\]
02

Solve for the Variable

Now, you need to isolate \(t\) completely. To do this, divide both sides of the equation by -16, which is the coefficient of \(t\): \[\frac{-16t}{-16} = \frac{64}{-16}\] Simplifying this gives: \[t = -4\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Solving Linear Equations
Solving linear equations is a fundamental concept in algebra. Linear equations are equations of the first degree, meaning they involve only the first power of the variable with no exponents. These equations are represented in the standard form:
  • ax + b = c
In the equation above,
  • - *a*, *b*, and *c* are constants, which are known numbers.
  • - *x* is the unknown variable that we need to solve for.
The goal of solving a linear equation is to find the value of the variable that makes the equation true. These equations are solvable through basic arithmetic operations, such as addition, subtraction, multiplication, and division. Practicing solving these types of equations strengthens understanding of mathematical principles and enhances problem-solving skills.
Isolation of Variables
Isolation of variables is a crucial step in solving equations. This process involves rearranging the equation to get the variable of interest by itself on one side of the equation. The strategy for isolating the variable is straightforward:
  • First, move any constants (numbers not multiplied or divided into the variable) to the opposite side of the equation. This is often accomplished using addition or subtraction.
  • Next, address the coefficient attached to the variable. This requires either dividing or multiplying both sides of the equation to leave the variable completely alone.
In the given exercise, the equation
  • -16t + 3 = 67
we first subtract 3 from both sides to handle the constant, simplifying the equation to
  • -16t = 64.
Then, we deal with the coefficient -16 by dividing both sides by -16 to solve for *t*. This focused method simplifies solving and enhances understanding of algebraic concepts.
Step-by-Step Solution
Breaking down a problem into a step-by-step solution helps in understanding and solving equations effectively. Here's why it's useful:
  • By tackling each step sequentially, learners can pay close attention to each part of the equation.
  • It can reduce errors by simplifying the problem-solving process.
For instance, let's look at the problem:
  • -16t + 3 = 67.
The first step is to isolate the term
  • -16t by getting rid of the constant +3, which we do by subtracting 3 from both sides, resulting in -16t = 64.
Next, tackle the coefficient -16 affecting *t* by dividing both sides by -16.
  • This gives the solution t = -4.
By outlining these steps clearly, students can follow the logic more easily, making algebraic problem-solving more approachable.

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Most popular questions from this chapter

Solve each equation. $$\frac{3}{4} x-\frac{2}{3}=\frac{5}{6}$$

Ginny has a collection of 425 coins consisting of pennies, nickels, and dimes. She has 50 more nickels than pennies and 25 more dimes than nickels. How many coins of each kind does she have?

Solve each equation. (a) \(-2(x-1)=-2 x+2\) (b) \(3(x+4)=3 x-4\) (c) \(5(x-1)=-5 x-5\) (d) \(\frac{x-3}{3}+4=3\) (e) \(\frac{x+2}{3}+1=\frac{x-2}{3}\) (f) \(\frac{x-1}{5}-2=\frac{x-11}{5}\) (g) \(4(x-2)-2(x+3)=2(x+6)\) (h) \(5(x+3)-3(x-5)=2(x+15)\) (i) \(7(x-1)+4(x-2)=15(x-1)\)

Find a number such that 20 more than one-third of the number equals three- fourths of the number.

Solve each equation. $$6(n+7)=8$$
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