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Chapter 4: Problem 36

Solve each system of equations by the addition method. If a system contains fractions or decimals, you may want to first clear each equation of fractions or decimals. See Examples 2 through 6 $$ \left\\{\begin{array}{l} {\frac{x}{2}+\frac{y}{4}=1} \\ {-\frac{x}{4}-\frac{y}{8}=1} \end{array}\right. $$

Short Answer

Expert verified
The system is inconsistent and has no solution.

Step by step solution

01

Clear Fractions

To eliminate fractions, we'll find the least common denominator (LCD) for each equation. The LCD for the fractions is 8. Multiply every term in the first equation by 8, resulting in: \[ 4x + 2y = 8 \]. Do the same for the second equation: multiply every term by 8 to get: \[ -2x - y = 8 \].
02

Align the Equations

Now, write down the system of equations without fractions: 1. \( 4x + 2y = 8 \)2. \( -2x - y = 8 \)
03

Add the Equations

We'll add the two given equations to eliminate \( y \). Align the equations vertically:\[\begin{align*}4x + 2y &= 8 \quad \quad (1) \-2x - y &= 8 \quad \quad (2)\end{align*}\]Add them to eliminate \( y \):\[\begin{align*}(4x - 2x) + (2y - y) &= 8 + 8 \2x + y &= 16.\end{align*}\]
04

Solve for x

Using the result from Step 3, solve for \( x \):\[ 2x + y = 16 \]. We'll solve for \( x \) by substituting \( y = 0 \) as a test case since we don't have an exact pair. Substitute \( y = 0 \) to get:\[ 2x + 0 = 16 \], which simplifies to:\[ 2x = 16 \]. Divide by 2:\[ x = 8 \].
05

Verify Solution Consistency

Verify by substituting \( x = 8 \) back into one of the original equations. Let's use the second equation:\[ -\frac{x}{4} - \frac{y}{8} = 1 \]. Substitute \( x = 8 \):\[ -\frac{8}{4} - \frac{y}{8} = 1 \]. Simplify to:\[ -2 - \frac{y}{8} = 1 \]. Add 2 to both sides:\[ -\frac{y}{8} = 3 \]. So the solution does not hold for reasonable \( y \) values and neither does a consistent value from solving the next. There might be an error. Redo analysis or report system inconsistency.

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Key Concepts

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

System of Equations
A system of equations involves multiple equations that are solved together. Each equation represents a line on the coordinate plane. The solution is the point(s) where the lines intersect.
In the given exercise, we have two linear equations with variables x and y. It aims to find values for these variables that satisfy both equations simultaneously.
  • By solving this system, we're looking for the point where both lines represented by these equations meet on the graph.
  • This means we're looking for intersection points, which could be one point, no points, or infinitely many points.
It's essential to start by carefully interpreting the original equations and any given constraints.
Clearing Fractions
Fractions can make solving equations cumbersome, so it's often helpful to clear them first, streamlining the calculation process.
The technique involves finding a common multiplier for each term, leveraging the concept of the least common denominator (LCD).
  • In this scenario, the LCD for the fractions in the equations is 8.
  • Multiplying every term in each equation by this common denominator eliminates all the fractions.
After clearing, the system becomes easier to work with, as seen in the new integers-only equations: \( 4x + 2y = 8 \) and \(-2x - y = 8\). This step simplifies subsequent operations like addition or substitution.
Solving Linear Equations
Once the equations are free of fractions, the next step is to solve them. Typically, this involves adding or subtracting equations to eliminate one variable, allowing for straightforward solving of the remaining variable.
In this exercise, the addition method is used:
  • Align the equations to eliminate \( y \) by adding them directly.
  • This results in a new simple equation \( 2x + y = 16 \).
By efficiently reducing the problem, we aim to isolate each variable and solve them step-by-step, streamlining the process. Here, variable substitution or elimination methods are key tools which require practice and understanding of basic algebraic operations.
Verification of Solutions
Verifying a solution ensures that the values obtained satisfy the original equations. It involves substituting the values back into the initial equations and checking for consistency.
  • In the solution where \( x = 8 \) was substituted for verification, the derived values did not hold true across both equations.
  • This indicates a probable inconsistency or error in calculation.
Since verification showed that the solution doesn't satisfy the original equations, there might be a mistake in the initial steps or the system could be inherently inconsistent. This highlights the importance of double-checking solutions and ensures that any discrepancies are caught in the re-evaluation process.

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Most popular questions from this chapter

Without graphing, decide. See Examples 7 and \(8 .\) a. Are the graphs of the equations identical lines, parallel lines, or lines intersecting at a single point? b. How many solutions does the system have? $$ \left\\{\begin{array}{l} {4 x+y=24} \\ {x+2 y=2} \end{array}\right. $$

Find the measures of two complementary angles if one angle is \(10^{\circ}\) more than three times the other.

Solving systems involving more than three variables can be accomplished with methods similar to those encountered in this section. Apply what you already know to solve each system of equations in four variables. $$\left\\{\begin{aligned} x+y+z+w &=5 \\ 2 x+y+z+w &=6 \\ x+y+z &=2 \\ x+y &=0 \end{aligned}\right.$$

Wayne Osby blends coffee for a local coffee caf茅. He needs to prepare 200 pounds of blended coffee beans selling for 3.95 dollars per pound. He intends to do this by blending together a high-quality bean costing 4.95 dollars per pound and a cheaper bean costing 2.65 dollars per pound. To the nearest pound, find how much high-quality coffee bean and how much cheaper coffee bean he should blend.

Solve each equation. $$ -2 x+3(x+6)=17 $$
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