Prove that any pair of linear equations with identical ratios a2a1 and b2b1 but different ratio c2c1 will never intersect on a Cartesian plane regardless of coordinate scale used (Rn).
Get the complete, step-by-step math solution for: "Prove that any pair of linear equations with identical ratios (a_1)/(a_2) and (b_1)/(b_2) but different ratio (c_1)/(c_2) will never intersect on a Ca...". Powered by SolveForX AI math tutor.
Step-by-Step Solution
Step 1: Represent the linear equations
We start by writing down the general form of two linear equations in two variables, x and y. These equations represent two straight lines on a Cartesian plane.
Step 2: Analyze the given conditions
The problem states that the ratio of the coefficients of x is equal to the ratio of the coefficients of y, but this ratio is not equal to the ratio of the constant terms. This is a key condition for the relationship between the two lines.
Step 3: Derive the relationship between slopes
We can find the slope of each line by rearranging the equations into the slope-intercept form (y=mx+c). The slope of a line Ax+By+C=0 is given by m=−BA.
Step 4: Compare the slopes
From the given condition a2a1=b2b1, we can rearrange it to show that b1a1=b2a2. This directly implies that the slopes of the two lines are equal (m1=m2). Lines with equal slopes are either parallel or coincident.
Step 5: Compare the y-intercepts
Now, let's consider the y -intercepts of the two lines. The y -intercept of a line Ax+By+C=0 is given by y=−BC (when x=0).
Step 6: Conclude non-intersection
Since b2b1=c2c1 and we know a2a1=b2b1, it implies that b1c1=b2c2. Therefore, the y -intercepts of the two lines are different (y1=y2). Since the lines have the same slope but different y -intercepts, they must be parallel and distinct. Parallel and distinct lines never intersect.