Get clear, concise, and accurate answers to your questions on IDNLearn.com. Find reliable solutions to your questions quickly and accurately with help from our dedicated community of experts.
Sagot :
To determine the nature of the system of linear equations [tex]\( 4x + 3y = 12 \)[/tex] and [tex]\( 2x + 9y = 15 \)[/tex], we need to analyze their solutions and the relationships between their coefficients.
### Step 1: Understanding the equations
We are given the following system of equations:
1. [tex]\( 4x + 3y = 12 \)[/tex]
2. [tex]\( 2x + 9y = 15 \)[/tex]
### Step 2: Solving the system using substitution or elimination
#### Using Substitution
Let's take the first equation and solve for [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex]:
[tex]\[ y = \frac{12 - 4x}{3} \][/tex]
Now, substitute [tex]\( y \)[/tex] in the second equation:
[tex]\[ 2x + 9\left(\frac{12 - 4x}{3}\right) = 15 \][/tex]
Simplify and solve for [tex]\( x \)[/tex]:
[tex]\[ 2x + 3(12 - 4x) = 15 \][/tex]
[tex]\[ 2x + 36 - 12x = 15 \][/tex]
[tex]\[ -10x + 36 = 15 \][/tex]
[tex]\[ -10x = 15 - 36 \][/tex]
[tex]\[ -10x = -21 \][/tex]
[tex]\[ x = \frac{21}{10} \][/tex]
Now substitute [tex]\( x = \frac{21}{10} \)[/tex] back into the expression for [tex]\( y \)[/tex]:
[tex]\[ y = \frac{12 - 4\left(\frac{21}{10}\right)}{3} \][/tex]
[tex]\[ y = \frac{12 - \frac{84}{10}}{3} \][/tex]
[tex]\[ y = \frac{12 - 8.4}{3} \][/tex]
[tex]\[ y = \frac{3.6}{3} \][/tex]
[tex]\[ y = \frac{6}{5} \][/tex]
So the solution to the system of equations is [tex]\( x = \frac{21}{10} \)[/tex] and [tex]\( y = \frac{6}{5} \)[/tex].
### Step 3: Determining the nature of the lines
To determine if the lines are intersecting, parallel, or coincident, we need to look at the coefficients:
First equation: [tex]\( 4x + 3y = 12 \)[/tex]
- Coefficients: [tex]\( a_1 = 4 \)[/tex], [tex]\( b_1 = 3 \)[/tex], [tex]\( c_1 = -12 \)[/tex]
Second equation: [tex]\( 2x + 9y = 15 \)[/tex]
- Coefficients: [tex]\( a_2 = 2 \)[/tex], [tex]\( b_2 = 9 \)[/tex], [tex]\( c_2 = -15 \)[/tex]
### Step 4: Compare the coefficients
The criterion to determine the relationship:
- If [tex]\( \frac{a_1}{a_2} \neq \frac{b_1}{b_2} \)[/tex], the lines are intersecting.
- If [tex]\( \frac{a_1}{a_2} = \frac{b_1}{b_2} \)[/tex] but [tex]\( \frac{a_1}{a_2} \neq \frac{c_1}{c_2} \)[/tex], the lines are parallel.
- If [tex]\( \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} \)[/tex], the lines are coincident.
Evaluating the ratios:
[tex]\[ \frac{a_1}{a_2} = \frac{4}{2} = 2 \][/tex]
[tex]\[ \frac{b_1}{b_2} = \frac{3}{9} = \frac{1}{3} \][/tex]
[tex]\[ \frac{c_1}{c_2} = \frac{-12}{-15} = \frac{4}{5} \][/tex]
Since [tex]\( \frac{a_1}{a_2} \neq \frac{b_1}{b_2} \)[/tex], the lines are intersecting.
### Conclusion
The system of equations [tex]\( 4x + 3y = 12 \)[/tex] and [tex]\( 2x + 9y = 15 \)[/tex] represents intersecting lines with the solution [tex]\( x = \frac{21}{10} \)[/tex] and [tex]\( y = \frac{6}{5} \)[/tex]. Thus, the lines are intersecting.
### Step 1: Understanding the equations
We are given the following system of equations:
1. [tex]\( 4x + 3y = 12 \)[/tex]
2. [tex]\( 2x + 9y = 15 \)[/tex]
### Step 2: Solving the system using substitution or elimination
#### Using Substitution
Let's take the first equation and solve for [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex]:
[tex]\[ y = \frac{12 - 4x}{3} \][/tex]
Now, substitute [tex]\( y \)[/tex] in the second equation:
[tex]\[ 2x + 9\left(\frac{12 - 4x}{3}\right) = 15 \][/tex]
Simplify and solve for [tex]\( x \)[/tex]:
[tex]\[ 2x + 3(12 - 4x) = 15 \][/tex]
[tex]\[ 2x + 36 - 12x = 15 \][/tex]
[tex]\[ -10x + 36 = 15 \][/tex]
[tex]\[ -10x = 15 - 36 \][/tex]
[tex]\[ -10x = -21 \][/tex]
[tex]\[ x = \frac{21}{10} \][/tex]
Now substitute [tex]\( x = \frac{21}{10} \)[/tex] back into the expression for [tex]\( y \)[/tex]:
[tex]\[ y = \frac{12 - 4\left(\frac{21}{10}\right)}{3} \][/tex]
[tex]\[ y = \frac{12 - \frac{84}{10}}{3} \][/tex]
[tex]\[ y = \frac{12 - 8.4}{3} \][/tex]
[tex]\[ y = \frac{3.6}{3} \][/tex]
[tex]\[ y = \frac{6}{5} \][/tex]
So the solution to the system of equations is [tex]\( x = \frac{21}{10} \)[/tex] and [tex]\( y = \frac{6}{5} \)[/tex].
### Step 3: Determining the nature of the lines
To determine if the lines are intersecting, parallel, or coincident, we need to look at the coefficients:
First equation: [tex]\( 4x + 3y = 12 \)[/tex]
- Coefficients: [tex]\( a_1 = 4 \)[/tex], [tex]\( b_1 = 3 \)[/tex], [tex]\( c_1 = -12 \)[/tex]
Second equation: [tex]\( 2x + 9y = 15 \)[/tex]
- Coefficients: [tex]\( a_2 = 2 \)[/tex], [tex]\( b_2 = 9 \)[/tex], [tex]\( c_2 = -15 \)[/tex]
### Step 4: Compare the coefficients
The criterion to determine the relationship:
- If [tex]\( \frac{a_1}{a_2} \neq \frac{b_1}{b_2} \)[/tex], the lines are intersecting.
- If [tex]\( \frac{a_1}{a_2} = \frac{b_1}{b_2} \)[/tex] but [tex]\( \frac{a_1}{a_2} \neq \frac{c_1}{c_2} \)[/tex], the lines are parallel.
- If [tex]\( \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} \)[/tex], the lines are coincident.
Evaluating the ratios:
[tex]\[ \frac{a_1}{a_2} = \frac{4}{2} = 2 \][/tex]
[tex]\[ \frac{b_1}{b_2} = \frac{3}{9} = \frac{1}{3} \][/tex]
[tex]\[ \frac{c_1}{c_2} = \frac{-12}{-15} = \frac{4}{5} \][/tex]
Since [tex]\( \frac{a_1}{a_2} \neq \frac{b_1}{b_2} \)[/tex], the lines are intersecting.
### Conclusion
The system of equations [tex]\( 4x + 3y = 12 \)[/tex] and [tex]\( 2x + 9y = 15 \)[/tex] represents intersecting lines with the solution [tex]\( x = \frac{21}{10} \)[/tex] and [tex]\( y = \frac{6}{5} \)[/tex]. Thus, the lines are intersecting.
Thank you for joining our conversation. Don't hesitate to return anytime to find answers to your questions. Let's continue sharing knowledge and experiences! IDNLearn.com is committed to your satisfaction. Thank you for visiting, and see you next time for more helpful answers.
