The solution set for the system of linear equations \(x + y = 8\) and \(y = 3x\) is \(\boxed{\left\{{\left({{\mathbf{2,6}}}\right)}\right\}}\).
Further explanation:
It is given that the system of linear equations are \(x + y = 8\) and \(y = 3x\).
Consider the given equations as follows:
\(x + y = 8\,\,\,\) ......(1)
\(y = 3x\,\,\,\) ......(2)
From equation (2), the value of \(y\) in terms of \(x\) is\(3x\).
Now, substitute \(3x\) for \(y\) in the equation (1) as follows:
\(x + 3x = 8\)
The variable is eliminated in the above equation.
Simplify the equation as follows:
\(\begin{aligned}x + 3x&=8\\4x&=8\\x&=\frac{8}{4}\\x&=2\\\end{aligned}\)
Therefore, the value of \(x\) is \(2\).
Substitute \(2\) for \(x\) in the equation (2) and obtain the value of \(y\) as shown below.
\(\begin{aligned}y&= 3\left( 2\right)\\&=6\\\end{aligned}\)
Therefore, the value of \(y\) is \(6\).
Thus, the ordered pair for the given system of linear equation is \(\left({{\mathbf{2,6}}} \right)\).
Check whether the obtained solution \((2,6)\) satisfies the given equations or not.
Substitute \(2\) for \(x\) and \(6\) for \(y\) in the equation (1) and check the equation.
\(\begin{aligned}2 + 6\mathop&_=^? 8\hfill \\\,\,\,\,\,\,\,8 &= 8\,\,\,\hfill\\\end{aligned}\) (True)
The ordered pair \(\left({2,6}\right)\) satisfies the equation (1).
Substitute \(2\) for \(x\) and \(6\) for \(y\) in the equation (2) and check the equation.
\(\begin{aligned}6\mathop&_= ^? \:3\left( 2 \right)\hfill\\6&= 6\,\,\,\,\,\,\,\,\,\,\,\,\hfill\\\end{aligned}\) (True)
The ordered pair \(\left({2,6}\right)\) satisfies the equation (2).
Thus, the solution set for the system of linear equations \(x + y = 8\) and \(y = 3x\) is \(\boxed{\left\{{\left({{\mathbf{2,6}}}\right)}\right\}}\).
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Answer Details:
Grade: Junior High School
Subject: Mathematics
Chapter: Linear equations
Keywords: Substitution, linear equation, system of linear equations in two variables, variables, mathematics,\(x + y = 8\) ,\(y = 3x\) , solution set
