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2x + 4y = 15
6x +12y = 45

What would the solution to the system of equations be?

Sagot :

Esther

Answer:

They both have an infinite number of solutions.

Step-by-step explanation:

Given system of equations:

a) 2x + 4y = 15

b) 6x + 12y = 45

Slope-intercept form: y = mx + b

where:

  • m is the slope
  • b is the y-intercept (when x = 0)

Rewrite both equations into slope-intercept form:

a) 2x + 4y = 15

⇒ 2x + 4y = 15 [subtract 2x from both sides]

⇒ 2x - 2x + 4y = 15 - 2x

⇒ 4y = - 2x + 15 [divide both sides by 4]

⇒ 4y ÷ 4 = (-2x ÷ 4) + (15 ÷ 4)

[tex]\sf \implies y = -\dfrac{1}{2}x\ + \dfrac{15}{4} \ or \ y=-0.5x\ + 3.75[/tex]

b) 6x + 12y = 45

⇒ 6x + 12y = 45 [subtract 6x from both sides]

⇒ 6x - 6x + 12y = 45 - 6x

⇒ 12y = - 6x + 45 [divide both sides by 12]

⇒ 12y ÷ 12 = (-6x ÷ 12) + (45 ÷ 12)

[tex]\sf \implies y = -\dfrac{1}{2}x\ + \dfrac{15}{4} \ or \ y=-0.5x\ + 3.75[/tex]

New equations:

[tex]\sf a)\ y = -\dfrac{1}{2}x\ + \dfrac{15}{4} \ or \ y=-0.5x\ + 3.75\\\\\sf b)\ y = -\dfrac{1}{2}x\ + \dfrac{15}{4} \ or \ y=-0.5x\ + 3.75[/tex]

Both equations have the same slope (-½), and y-intercept (3.75). Therefore, they both have an infinite number of solutions.

System of equations can have the following:

No Solution: the same slope (both lines will be parallel)

One Solution: different slopes and different y-intercepts

Infinitely Many Solutions: the same slope and y-intercept

Learn more about system of equations here:

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igoroleshko156

Looking at the given expression, they do not seem to be in the slope-intercept form which is the most common form used for linear expression. Let us convert the equations that were given in the problem statement to follow the slope-intercept form.

  • Slope-Intercept Form ⇒ [tex]y = mx + b[/tex]
  • m = slope
  • b = y-intercept

Equation #1

Subtract 2x from both sides

  • [tex]2x + 4y = 15[/tex]
  • [tex]2x - 2x + 4y = 15 - 2x[/tex]
  • [tex]4y = -2x + 15[/tex]

Divide both sides by 4

  • [tex]\frac{4y}{4} = \frac{-2}{4}x + \frac{15}{4}[/tex]
  • [tex]y = \frac{-2}{4}x + \frac{15}{4}[/tex]
  • [tex]y = -0.5x + 3.75[/tex]

Equation #2

Subtract 6x from both sides

  • [tex]6x + 12y = 45[/tex]
  • [tex]6x - 6x + 12y = 45 - 6x[/tex]
  • [tex]12y = -6x + 45[/tex]

Divide both sides by 12

  • [tex]\frac{12y}{12} = \frac{-6}{12}x + \frac{45}{12}[/tex]
  • [tex]y = \frac{-6}{12}x + \frac{45}{12}[/tex]
  • [tex]y = -0.5x + 3.75[/tex]

Since both the first and second equation have the same exact number which means that they will fall exactly on top of each other.  Therefore, there are infinite solutions as they will always continue on top of each other.

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