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Sagot :
To solve the system of linear equations, we need to find the values of [tex]\(x\)[/tex] and [tex]\(y\)[/tex] that satisfy both equations simultaneously. The given system of equations is:
[tex]\[ \begin{cases} y = -2x + 7 \\ y = 5x - 7 \end{cases} \][/tex]
Here's the step-by-step solution:
1. Set the equations equal to each other since both are equal to [tex]\(y\)[/tex]:
[tex]\[ -2x + 7 = 5x - 7 \][/tex]
2. Combine like terms to isolate [tex]\(x\)[/tex]:
[tex]\[ 7 + 7 = 5x + 2x \][/tex]
[tex]\[ 14 = 7x \][/tex]
3. Solve for [tex]\(x\)[/tex]:
[tex]\[ x = \frac{14}{7} \][/tex]
[tex]\[ x = 2 \][/tex]
Now that we have determined [tex]\(x = 2\)[/tex], we need to find the corresponding value of [tex]\(y\)[/tex].
4. Substitute [tex]\(x = 2\)[/tex] back into one of the original equations. We'll use the first equation:
[tex]\[ y = -2(2) + 7 \][/tex]
[tex]\[ y = -4 + 7 \][/tex]
[tex]\[ y = 3 \][/tex]
Therefore, the solution to the system of equations is:
[tex]\[ \begin{cases} x = 2 \\ y = 3 \end{cases} \][/tex]
So, the solution is [tex]\( \boxed{(2, 3)} \)[/tex].
[tex]\[ \begin{cases} y = -2x + 7 \\ y = 5x - 7 \end{cases} \][/tex]
Here's the step-by-step solution:
1. Set the equations equal to each other since both are equal to [tex]\(y\)[/tex]:
[tex]\[ -2x + 7 = 5x - 7 \][/tex]
2. Combine like terms to isolate [tex]\(x\)[/tex]:
[tex]\[ 7 + 7 = 5x + 2x \][/tex]
[tex]\[ 14 = 7x \][/tex]
3. Solve for [tex]\(x\)[/tex]:
[tex]\[ x = \frac{14}{7} \][/tex]
[tex]\[ x = 2 \][/tex]
Now that we have determined [tex]\(x = 2\)[/tex], we need to find the corresponding value of [tex]\(y\)[/tex].
4. Substitute [tex]\(x = 2\)[/tex] back into one of the original equations. We'll use the first equation:
[tex]\[ y = -2(2) + 7 \][/tex]
[tex]\[ y = -4 + 7 \][/tex]
[tex]\[ y = 3 \][/tex]
Therefore, the solution to the system of equations is:
[tex]\[ \begin{cases} x = 2 \\ y = 3 \end{cases} \][/tex]
So, the solution is [tex]\( \boxed{(2, 3)} \)[/tex].
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