Welcome to our math problem solving session! Today we'll work through two mathematical problems step by step. I'll use clear visual demonstrations to help you understand the concepts and find the solutions. Let's begin by examining each problem carefully and developing a systematic approach to solve them.
Let's solve our first problem, which is a quadratic equation: 2x squared plus 5x minus 3 equals zero. We'll use the quadratic formula where x equals negative b plus or minus the square root of b squared minus 4ac, all divided by 2a. Here, a equals 2, b equals 5, and c equals negative 3. First, we calculate the discriminant: b squared minus 4ac equals 25 plus 24, which equals 49. Since the discriminant is positive, we have two real solutions. The first solution is x equals negative 5 plus 7, divided by 4, which equals one half. The second solution is x equals negative 5 minus 7, divided by 4, which equals negative 3.
Now let's verify our solutions graphically by plotting the parabola y equals 2x squared plus 5x minus 3. The graph shows a parabola opening upward, and we can see that it crosses the x-axis at exactly two points. These x-intercepts occur at x equals one half and x equals negative 3, which perfectly match the roots we calculated using the quadratic formula. This graphical verification confirms that our algebraic solution is correct.
Now let's solve our second problem, which is a system of linear equations. We have 3x plus 2y equals 12, and x minus y equals 1. We'll use the substitution method. From the second equation, we can express x as y plus 1. Now we substitute this into the first equation: 3 times y plus 1, plus 2y equals 12. Expanding this gives us 3y plus 3 plus 2y equals 12. Combining like terms, we get 5y equals 9, so y equals 9 fifths. Substituting back, x equals 9 fifths plus 1, which equals 14 fifths. Therefore, our solution is x equals 14 fifths and y equals 9 fifths.
Finally, let's verify our solution graphically by plotting both equations as lines on a coordinate system. The blue line represents 3x plus 2y equals 12, and the red line represents x minus y equals 1. These two lines intersect at exactly one point, which represents the solution to our system. The intersection occurs at the coordinates 14 fifths, 9 fifths, which equals 2.8, 1.8. This perfectly matches our algebraic solution, confirming that we solved both problems correctly. The graphical method provides an excellent visual verification of our mathematical work.