To solve the system of equations given by:
[tex]\[ -x + y = -3.5 \][/tex]
[tex]\[ x + 3y = 9.5 \][/tex]
we need to find the values of [tex]\(x\)[/tex] and [tex]\(y\)[/tex] that satisfy both equations simultaneously. Here’s a detailed, step-by-step solution:
1. Step 1: Solve the first equation for [tex]\(y\)[/tex] in terms of [tex]\(x\)[/tex]:
The first equation is:
[tex]\[ -x + y = -3.5 \][/tex]
To isolate [tex]\(y\)[/tex], add [tex]\(x\)[/tex] to both sides:
[tex]\[ y = x - 3.5 \][/tex]
2. Step 2: Substitute [tex]\(y = x - 3.5\)[/tex] into the second equation:
The second equation is:
[tex]\[ x + 3y = 9.5 \][/tex]
Substitute [tex]\(y\)[/tex] from the first equation:
[tex]\[ x + 3(x - 3.5) = 9.5 \][/tex]
3. Step 3: Simplify and solve for [tex]\(x\)[/tex]:
Expand the equation:
[tex]\[ x + 3x - 10.5 = 9.5 \][/tex]
Combine like terms:
[tex]\[ 4x - 10.5 = 9.5 \][/tex]
Add 10.5 to both sides to isolate the term with [tex]\(x\)[/tex]:
[tex]\[ 4x = 20 \][/tex]
Divide both sides by 4:
[tex]\[ x = 5 \][/tex]
4. Step 4: Substitute [tex]\(x = 5\)[/tex] back into the equation [tex]\(y = x - 3.5\)[/tex]:
Use the expression for [tex]\(y\)[/tex]:
[tex]\[ y = 5 - 3.5 \][/tex]
[tex]\[ y = 1.5 \][/tex]
Therefore, the solution to the system of equations is:
[tex]\[ x = 5 \][/tex]
[tex]\[ y = 1.5 \][/tex]
So, the values of [tex]\(x\)[/tex] and [tex]\(y\)[/tex] that satisfy the system of equations [tex]\(-x + y = -3.5\)[/tex] and [tex]\(x + 3y = 9.5\)[/tex] are [tex]\( x = 5 \)[/tex] and [tex]\( y = 1.5 \)[/tex].
