Graph the equations of each system on the Cartesian plane. Then, determine its solution by applying any method.
a. {3s+ 4t = 15
{ 2s+t=5
Answer (1)
To solve the system of equations given by { 3 s + 4 t = 15 } and { 2 s + t = 5 } , we first graph the equations on the Cartesian plane and then find their solution algebraically.
Step 1: Graphing the Equations
Equation 1: 3 s + 4 t = 15
To find the x-intercept, set t = 0 : 3 s + 4 ( 0 ) = 15 ⟹ 3 s = 15 ⟹ s = 5
To find the y-intercept, set s = 0 : 3 ( 0 ) + 4 t = 15 ⟹ 4 t = 15 ⟹ t = 4 15
Plot the points ( 5 , 0 ) and ( 0 , 4 15 ) , then draw the line.
Equation 2: 2 s + t = 5
To find the x-intercept, set t = 0 : 2 s + 0 = 5 ⟹ 2 s = 5 ⟹ s = 2 5
To find the y-intercept, set s = 0 : 2 ( 0 ) + t = 5 ⟹ t = 5
Plot the points ( 2 5 , 0 ) and ( 0 , 5 ) , then draw the line.
Step 2: Solving the System Algebraically (by Substitution or Elimination)
Let's use the elimination method:
Multiply the second equation by 4 to eliminate t : 4 ( 2 s + t ) = 4 ( 5 ) ⟹ 8 s + 4 t = 20
Subtract the first equation from this new equation: ( 8 s + 4 t ) − ( 3 s + 4 t ) = 20 − 15 ⟹ 5 s = 5
Solving for s : 5 s = 5 ⟹ s = 1
Substitute s = 1 into the second original equation: 2 ( 1 ) + t = 5 ⟹ 2 + t = 5 ⟹ t = 3
Solution: The solution of the system of equations is s = 1 and t = 3 . Therefore, the intersection point of the two lines on the graph is ( 1 , 3 ) .
Graphically, the lines 3 s + 4 t = 15 and 2 s + t = 5 intersect at the point ( 1 , 3 ) , which verifies our algebraic solution.
