Certainly! Let's solve the simultaneous equations step by step:
Given two equations:
1. [tex]\( y = -2x + 5 \)[/tex]
2. [tex]\( y = \frac{1}{2}x + 1 \)[/tex]
To find the values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] that satisfy both equations simultaneously, we set the equations equal to each other, since both are equal to [tex]\( y \)[/tex]:
[tex]\[ -2x + 5 = \frac{1}{2}x + 1 \][/tex]
Next, let's clear the fraction by multiplying every term by 2:
[tex]\[ 2(-2x + 5) = 2 \left(\frac{1}{2}x + 1\right) \][/tex]
This simplifies to:
[tex]\[ -4x + 10 = x + 2 \][/tex]
Now, we'll solve for [tex]\( x \)[/tex]:
First, isolate [tex]\( x \)[/tex] on one side of the equation. We can do this by subtracting [tex]\( x \)[/tex] and 10 from both sides:
[tex]\[ -4x - x + 10 - 10 = x - x + 2 - 10 \][/tex]
This simplifies to:
[tex]\[ -5x = -8 \][/tex]
Now, solve for [tex]\( x \)[/tex] by dividing both sides by -5:
[tex]\[ x = \frac{-8}{-5} \][/tex]
[tex]\[ x = \frac{8}{5} \][/tex]
Once we have [tex]\( x \)[/tex], we'll substitute it back into one of the original equations to find [tex]\( y \)[/tex]. Let's use the first equation:
[tex]\[ y = -2x + 5 \][/tex]
Substitute [tex]\( x = \frac{8}{5} \)[/tex]:
[tex]\[ y = -2 \left(\frac{8}{5}\right) + 5 \][/tex]
Simplify this:
[tex]\[ y = -\frac{16}{5} + 5 \][/tex]
To combine the terms, convert 5 into a fraction with a denominator of 5:
[tex]\[ y = -\frac{16}{5} + \frac{25}{5} \][/tex]
This simplifies to:
[tex]\[ y = \frac{-16 + 25}{5} \][/tex]
[tex]\[ y = \frac{9}{5} \][/tex]
Therefore, the solution to the system of equations is:
[tex]\[ x = \frac{8}{5}, \quad y = \frac{9}{5} \][/tex]
These values satisfy both original equations.
