Respuesta :
Answer:
Examine the system of equations.
–2x + 3y = 6
–4x + 6y = 12
Answer the questions to determine the number of solutions to the system of equations.
What is the slope of the first line?
✔ 2/3
What is the slope of the second line?
✔ 2/3
What is the y-intercept of the first line?
✔ 2
What is the y-intercept of the second line?
✔ 2
How many solutions does the system have?
✔ infinitely many
The equations are a multiple of the other, therefore, by the multiplicative
property of equality, the equations are equivalent.
Response:
- The slope and y-intercept of the first equation are [tex]\underline{\dfrac{2}{3} \ and \ 2}[/tex] respectively
- The slope and y-intercept of the second equation are [tex]\underline{\dfrac{2}{3} \, and \, 2}[/tex]
- The system of equations have infinitely many solutions.
Methods used to obtain the above response.
The given system of equations are;
-2·x + 3·y = 6
-4·x + 6·y = 12
Required:
The slope of the first line.
Solution:
The slope of the first line is given by the coefficient of x when the equation is expressed in the form; y = m·x + c.
Therefore, from -2·x + 3·y = 6, we have;
3·y = 2·x + 6
[tex]y = \dfrac{2}{3} \cdot x + \dfrac{6}{3} = \dfrac{2}{3} \cdot x + 2[/tex]
[tex]y =\dfrac{2}{3} \cdot x + 2[/tex]
[tex]\underline{The \ slope \ of \ the \ first \ equation \ is \ \dfrac{2}{3}}[/tex]
Required:
The slope of the second line;
Solution:
The equation of the second line, -4·x + 6·y = 12, can be expressed in the form;
[tex]y =\dfrac{4}{6} \cdot x + \dfrac{12}{6} = \dfrac{2}{3} \cdot x + 2[/tex]
[tex]y = \mathbf{\dfrac{2}{3} \cdot x + 2}[/tex]
[tex]\underline{The \ slope \ of \ the \ second \ equation \ is \ therefore \ \dfrac{2}{3}}[/tex]
- The y-intercept of the first line = 2
- The y-intercept of the second line = 2
Given that the equation have the same slope and the same y-intercept, the equations are equations of the same line, therefore;
- The equations have an infinite number of solutions
Learn more about the solutions of a system of equations here:
https://brainly.com/question/15356519
