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Chapter 4: Problem 15

Solve system using Cramer's rule. \(x-y=-4\) \(2 y=12\)

Short Answer

Expert verified
The solution is \( x = 2 \) and \( y = 6 \).

Step by step solution

01

Identify the System of Equations

Given system of equations: 1. \( x - y = -4 \) 2. \( 2y = 12 \) We need to solve this system using Cramer's rule.
02

Solve for One Variable

First solve one of the equations for one of the variables. Let's solve the second equation for \( y \): \( 2y = 12 \) \( y = 6 \)
03

Substitute and Solve for the Other Variable

Substitute \( y = 6 \) into the first equation: \( x - 6 = -4 \) Solve for \( x \): \( x = 2 \)
04

Check the Solution

Check the solution by substituting \( x = 2 \) and \( y = 6 \) back into the original equations: 1. \( 2 - 6 = -4 \) 2. \( 2 \times 6 = 12 \) Both equations are satisfied.

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Key Concepts

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

Understanding the System of Equations
A system of equations consists of multiple algebraic expressions collectively used to find the values of variables that satisfy all given equations simultaneously. Imagine two roads crossing at a point; that intersection point is the solution to our system. In this example, we are given two equations involving variables x and y:
1. \( x - y = -4 \)
2. \( 2y = 12 \)
These equations are called a 'system' because we need to find values of x and y that make both equations true at the same time.
Algebraic Solution via Cramer's Rule
Cramer's rule provides a straightforward method for solving a system of linear equations using determinants. However, in this particular case, we can solve the system algebraically because it already suggests an easy path:
1. First, identify the equations, which have already been done for us:
\( x - y = -4 \) and \( 2y = 12 \)
2. Solve one equation to isolate a variable. Here, solving \( 2y = 12 \) for y is simple:
\( y = 6 \)
Simple steps but essential to starting the algebraic solution correctly.
Solving for Variables
Once we have isolated y, we substitute its value back into another equation to find x. Here鈥檚 how this is done specifically:
1. Substitute \( y = 6 \) into \( x - y = -4 \):
\( x - 6 = -4 \)
2. Solve for x by isolating it:
\( x = 2 \)
To ensure our solution is correct, we substitute both x and y back into the original equations to check if they hold true:
1. Substituting into \( x - y = -4 \):
\( 2 - 6 = -4 \)
2. Substituting into \( 2y = 12 \):
\( 2 \times 6 = 12 \)
Both equations are satisfied, confirming that our solutions are correct.

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Most popular questions from this chapter

Write a step-by-step procedure for solving any system of two linear equations in two variables by the Gauss-Jordan elimination method. Have a classmate evaluate your procedure by using it to solve a system.

One acute angle of a right triangle is \(3^{\circ}\) more than twice the other acute angle. What are the sizes of the acute angles?

Solve each system using the Gauss-Jordan elimination method. $$ \begin{aligned} x-y-z &=0 \\ -x-y+2 z &=-1 \\ -x+y-2 z &=-3 \end{aligned} $$

Evaluate the determinant of each \(3 \times 3\) matrix using expansion by minors about the row or column of your choice. \(\left[\begin{array}{rrr}-2 & 0 & 1 \\ -3 & 2 & -5 \\ 4 & -2 & 6\end{array}\right]\)

Solve each problem by using a system of three equations in three unknowns. Salvador's Fruit Mart sells variety packs. The small pack contains three bananas, two apples, and one orange for 1.80 dollars . The medium pack contains four bananas, three apples, and three oranges for 3.05 dollars. The family size contains six bananas, five apples, and four oranges for 4.65 dollars . What price should Salvador charge for his lunch-box special that consists of one banana, one apple, and one orange?
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