Math Problem Statement

Find the point of intersection of two straight lines in each of the following pairs of equations: a) y = 2x - 1 and y = 3x - 3, b) x + y = 16 and x + 2y = 10, c) 2y = x - 5 and y = -2x + 5, d) 3y = 6 - 4x and 6y = x - 12

Solution

Let's solve the problem step by step. The question asks to find the point of intersection of two straight lines for each given pair of equations. We will go through each pair of equations:

a) y=2x1y = 2x - 1 and y=3x3y = 3x - 3

To find the point of intersection, set the two equations equal to each other:

2x1=3x32x - 1 = 3x - 3

Solving for xx:

1+3=3x2x-1 + 3 = 3x - 2x

x=2x = 2

Substitute x=2x = 2 back into one of the original equations, say y=2x1y = 2x - 1:

y=2(2)1=41=3y = 2(2) - 1 = 4 - 1 = 3

So, the point of intersection is (2,3)(2, 3).

b) x+y=16x + y = 16 and x+2y=10x + 2y = 10

We can solve this system by substitution or elimination. Let's use the elimination method. Subtract the second equation from the first:

(x+y)(x+2y)=1610(x + y) - (x + 2y) = 16 - 10

x+y=6-x + y = 6

This simplifies

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Math Problem Analysis

Mathematical Concepts

Algebra
Linear Equations
System of Equations
Intersection of Lines

Formulas

Set y = y for two equations: y1 = y2
Substitution method: substitute one equation into the other
Elimination method: subtract or add the equations to eliminate a variable

Theorems

Properties of Linear Systems
Point of Intersection

Suitable Grade Level

Grades 8-10