/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 66 Suppose \(4 x+3 y=12 .\) Find \(... [FREE SOLUTION] | 91影视

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Chapter 4: Problem 66

Suppose \(4 x+3 y=12 .\) Find \(x\) if: $$y=\frac{5}{3}$$

Short Answer

Expert verified
The value of \(x\) is \(\frac{7}{4}\).

Step by step solution

01

Substitute the given value of y into the equation

The original equation is \(4x + 3y = 12\). We are given that \(y = \frac{5}{3}\). Substitute this value into the equation to replace \(y\). The equation becomes: \(4x + 3\left(\frac{5}{3}\right) = 12\).
02

Simplify the equation

Simplify the terms inside the equation. Multiply the constants: \(3 \times \frac{5}{3} = 5\). The equation now becomes \(4x + 5 = 12\).
03

Isolate the variable x

To find the value of \(x\), isolate it on one side of the equation. Subtract 5 from both sides of the equation: \(4x + 5 - 5 = 12 - 5\). This simplifies to \(4x = 7\).
04

Solve for x

Divide both sides of the equation by 4 to solve for \(x\): \(x = \frac{7}{4}\).

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Key Concepts

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

Substitution Method
The substitution method is a fundamental technique used in solving linear equations, and it's especially useful when we know the value of one variable. In simple terms, substitution involves replacing a variable with a known value to simplify the equation. This is exactly what happened in our exercise where we knew the value of \( y \) and substituted it into the equation.
  • Identify the variable with a known value, in this case \( y = \frac{5}{3} \).
  • Replace the variable in the equation with this known value: \( 4x + 3 \left( \frac{5}{3} \right) = 12 \).
By replacing \( y \) with its known value, we transform our equation into one with a single variable \( x \) that we can now solve with greater ease.
Simplifying Equations
Simplifying equations involves reducing them to a form that is easier to solve. This often includes performing arithmetic operations to simplify terms. In our example, after substituting the value of \( y \), the equation became \( 4x + 3 \left( \frac{5}{3} \right) = 12 \). The next step was to simplify the expression within the equation.
When you see an expression like \( 3 \left( \frac{5}{3} \right) \), you should:
  • Multiply the coefficient outside the parentheses by the fractional value inside: \( 3 \times \frac{5}{3} = 5 \).
  • Simplify the equation by replacing \( 3 \left( \frac{5}{3} \right) \) with 5, resulting in a simpler equation of \( 4x + 5 = 12 \).
These steps make the equation much simpler and bring you closer to isolating the variable.
Isolating Variables
Isolating the variable is the process of manipulating the equation to have the variable on one side and everything else on the other. Once we've simplified our equation, \( 4x + 5 = 12 \), the goal becomes getting \( x \) by itself.
To isolate \( x \):
  • Subtract 5 from both sides of the equation to get \( 4x = 12 - 5 \), which simplifies to \( 4x = 7 \).
  • Divide both sides of the equation by 4 to solve for \( x \): \( x = \frac{7}{4} \).
By isolating \( x \), we've effectively solved the equation and found that \( x = \frac{7}{4} \). This method works the same way for any linear equation, allowing you to find the value of the unknown variable.

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Most popular questions from this chapter

The following equations contain parentheses. Apply the distributive property to remove the parentheses, then simplify each side before using the addition property of equality. $$5(2 a+1)-9 a=8-6$$

Simplify. $$\frac{\frac{5}{7}}{\frac{6}{7}}$$

Vehicle Weight If you can measure the area that the tires on your car contact the ground, and you know the air pressure in the tires, then you can estimate the weight of your car, in pounds, with the following formula: $$W=A P N$$ where \(W\) is the vehicle's weight in pounds, \(A\) is the average tire contact area with a hard surface in square inches, \(P\) is the air pressure in the tires in pounds per square inch (psi, or \(\mathrm{lb} / \mathrm{in}^{2}\) ), and \(N\) is the number of tires.(IMAGE CANNOT COPY) a. What is the approximate weight of a car if the average tire contact area is a rectangle 6 inches by 5 inches and if the air pressure in the tires is 30 psi?

Multiply. $$-\frac{1}{3}(-3)$$

On a certain day, the temperature on the ground is \(72^{\circ}\) Fahrenheit, and the temperature \(T\) at an altitude of \(A\) feet above the ground is given by the equation \(T=72-\frac{1}{300} A\). Graph this equation on the rectangular coordinate system here. Note that we have restricted our graph to positive values of \(A\) only. (GRAPH CANT COPY)
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