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Chapter 4: Problem 25

(a) solve by substitution. (b) if there is one solution, check. $$ \begin{aligned} &y=\frac{3}{4} x-2 \\ &y=-x+5 \end{aligned} $$

Short Answer

Expert verified
The solution is \[ x = 4 \] and \[ y = 1 \].

Step by step solution

01

Set Equations Equal

Since both equations are equal to y, set \[ \frac{3}{4} x - 2 \] equal to \[ -x + 5 \]. That gives: \[ \frac{3}{4} x - 2 = -x + 5 \]
02

Isolate Variable x

To solve for x, first eliminate the fractions by multiplying through by 4. \[ 4 \left( \frac{3}{4} x - 2 \right) = 4 \left( -x + 5 \right) \] which simplifies to \[ 3x - 8 = -4x + 20 \].
03

Combine Like Terms

Add 4x to both sides: \[ 3x + 4x - 8 = 20 \] which simplifies to \[ 7x - 8 = 20 \].
04

Solve for x

Add 8 to both sides to isolate the term with x: \[ 7x - 8 + 8 = 20 + 8 \] giving \[ 7x = 28 \]. Next, divide by 7: \[ x = 4 \].
05

Substitute x Back to Find y

Substitute the value of x (which is 4) back into the first equation: \[ y = \frac{3}{4}(4) - 2 \] which simplifies to \[ y = 3 - 2 = 1 \]. Thus, \[ y = 1 \].
06

Validate the Solution

Substitute \[ x = 4 \] into the second equation to check: \[ y = -4 + 5 \], which also gives \[ y = 1 \]. Both equations are satisfied, confirming the solution is correct.

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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 powerful technique used to solve systems of equations. It involves replacing one variable with an expression derived from another equation. In our exercise, since both equations are given in terms of y, we set them equal to each other.
This transforms the problem into a single equation with one variable.
This method is particularly useful when one equation is already solved for a variable.
Isolation of Variables
To isolate a variable means to get that variable alone on one side of the equation.
In our solution, we first set the equations equal since both were expressed as y. Then, we focused on isolating x.
By clearing fractions, using the technique of multiplying through by 4, we simplify the equation.
The goal is to arrange terms so all x are on one side.
This step prepares the equation for further simplification and solving.
Combining Like Terms
Combining like terms is crucial in simplifying an equation.
Once we isolated the variable and cleared fractions, our next step was to gather all x terms together.
Adding 4x to both sides gave us a straightforward equation with x terms combined.
This step ensures that equations are simplified and forms a groundwork for solving for the variable.
Validation of Solution
After solving for the variables, it's important to validate the solution.
This means substituting the found values back into the original equations to verify they satisfy both.
In our exercise, we substituted x = 4 back into both equations.
Both equations were satisfied, confirming the solution.
Validation is a crucial step ensuring accuracy.

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Most popular questions from this chapter

For exercises 1-40, use the five steps and a system of equations. Drink A is \(15 \%\) orange juice. Drink B is \(4 \%\) orange juice. Find the amount of each drink needed to make \(800 \mathrm{gal}\) of a new drink that is \(7 \%\) orange juice. Round to the nearest whole number.

A timber bolt with a diameter of \(\frac{5}{8}\) in. weighs \(1.2 \mathrm{lb}\) and costs $$\$ 3.65$$. A hex bolt with a diameter of 1 in. weighs \(3 \mathrm{lb}\) and costs $$\$ 7.82$$. The total weight of 20 bolts is \(51 \mathrm{lb}\). Find the number of timber bolts and the number of hex bolts.

For exercises \(85-88\), the completed problem has one mistake. (a) Describe the mistake in words, or copy down the whole problem and highlight or circle the mistake. (b) Do the problem correctly. Problem: Write a system of equations that represents the relationship of the variables. Each pound of Cereal A contains \(0.5 \mathrm{oz}\) of almonds and \(2 \mathrm{oz}\) of dried cranberries. Each pound of Cereal B contains 2 oz of almonds and 1 oz of dried cranberries. Find the amount of each cereal needed to make 400 lb of a new cereal that contains \(0.75 \mathrm{oz}\) of almonds per pound of cereal. (1 \(\mathrm{lb}=16 \mathrm{oz}\).) $$ \begin{aligned} &x=\text { amount of Cereal A } \\ &y=\text { amount of Cereal B } \end{aligned} $$ Incorrect Answer: \(x+y=400 \mathrm{lb}\) $$ \left(\frac{0.5 \mathrm{oz}}{1 \mathrm{lb}}\right) x+\left(\frac{2 \mathrm{oz}}{1 \mathrm{lb}}\right) y=0.75 \mathrm{oz} $$

A tugboat leaves a port pushing two barges, traveling at an average speed of \(6 \mathrm{mi}\) per hour. Four hours later, a tugboat without barges leaves the port, traveling at an average speed of \(8 \mathrm{mi}\) per hour. Find the time after the fast tugboat leaves port needed for the fast tugboat to catch up with the slower tugboat. Find the distance that the boats travel.

The cost of playground grass seed mix is $$\$ 2.16$$ per pound, and the cost of shade grass seed mix is $$\$ 3.60$$ per pound. If a total of \(500 \mathrm{lb}\) of grass seed is sold for $$\$ 1728$$, find the amount of playground grass seed mix and the amount of shade grass seed mix sold.
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