The solution set for the system of linear equations [tex]5x + 3y = 16[/tex] and [tex]x + 3y = 8[/tex]is[tex]\boxed{\left\{ {\left( {{\mathbf{2,2}}} \right)} \right\}}[/tex].
Further explanation:
It is given that the system of linear equations are [tex]5x + 3y = 16[/tex]and[tex]x + 3y = 8[/tex].
Consider the given equations as follows:
[tex]\begin{aligned}5x+3y=16\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,......\left(1\right)\hfill\\x+3y=8\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,......\left(2\right)\hfill\\\end{aligned}[/tex]
Isolate the variable [tex]y[/tex] in terms of [tex]x[/tex] from equation (1) as follows:
[tex]\begin{aligned}5x+3y&=16\\3y&=16-5x\\y&=\frac{{16-5x}}{3}\\\end{aligned}[/tex]
Therefore, the value of [tex]y[/tex] in terms of [tex]x[/tex] is[tex]\frac{{16-5x}}{3}[/tex].
Now, substitute [tex]\frac{{16-5x}}{3}[/tex] for [tex]y[/tex] in the equation (2) as follows:
[tex]x+3\left({\frac{{16-5x}}{3}}\right)=8[/tex]
The variable is eliminated in the above equation.
Simplify the equation as follows:
[tex]\begin{aligned}x+3\left({\frac{{16-5x}}{3}}\right)&=8\\x+16-5x&=8\\-4x+16&=8\\-4x&=8-16\\\end{aligned}[/tex]
Further simplify theequation.
[tex]\begin{aligned}-4x&=-8\\x&=\frac{8}{4}\\x&=2\\\end{aligned}[/tex]
Therefore, the value of [tex]x[/tex]is [tex]{\mathbf{2}}[/tex].
Substitute [tex]2[/tex] for [tex]x[/tex] in the equation (1) and obtain the value of [tex]y[/tex] as shown below.
[tex]\begin{aligned}5\left(2\right)+3y&=16\\10+3y&=16\\3y&=16-10\\3y&=6\\\end{aligned}[/tex]
Further simplify the above equation.
[tex]\begin{aligned}y&=\frac{6}{3}\\&=2\\\end{aligned}[/tex]
Therefore, the value of [tex]y[/tex]is [tex]{\mathbf{2}}[/tex].
Thus, the ordered pair for the system of linear equation is [tex]\left( {{\mathbf{2,2}}} \right)[/tex].
Check whether the obtained solution [tex]\left( {2,2} \right)[/tex] satisfies the given equations or not.
Substitute [tex]2[/tex] for [tex]x[/tex] and [tex]2[/tex] for [tex]y[/tex] in the equation (1) and check the equation.
[tex]\begin{aligned}5\left(2\right)+3\left(2\right)\mathop&=\limits^?16\hfill\\\,\,\,\,\,\,\,\,\,\,\,\,10+6\mathop&=\limits^?16\hfill\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,16&=16\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left( {{\text{True}}}\right)\hfill\\\end{aligned}[/tex]
The ordered pair [tex]\left( {2,2} \right)[/tex] satisfies the equation (1).
Substitute [tex]2[/tex] for [tex]x[/tex] and [tex]2[/tex] for [tex]y[/tex] in the equation (2) and check the equation.
[tex]\begin{aligned}2+3\left(2\right)\mathop&=\limits^?8\hfill\\\,\,\,\,\,\,\,2+6\mathop&=\limits^? 8\hfill\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,8&=8\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left({{\text{True}}}\right)\hfill\\\end{aligned}[/tex]
The ordered pair [tex]\left( {2,2} \right)[/tex] satisfies the equation (2).
Thus, the solution set for the system of linear equations [tex]5x + 3y = 16[/tex] and [tex]x + 3y = 8[/tex]is[tex]\boxed{\left\{ {\left( {{\mathbf{2,2}}} \right)} \right\}}[/tex].
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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,[tex]5x + 3y = 16[/tex],[tex]x + 3y = 8[/tex]
